Two-dimensional coordinate systems

Two-Dimensional Vectors

Introduction

In most mathematics courses up until this point, we deal with scalars. These are quantities which only need one number to express. For instance, the amount of gasoline used to drive to the grocery store is a scalar quantity because it only needs one number: 2 gallons.

In this unit, we deal with vectors. A vector is a directed line segment -- that is, a line segment that points one direction or the other. As such, it has an initial point and a terminal point. The vector starts at the initial point and ends at the terminal point, and the vector points towards the terminal point. A vector is drawn as a line segment with an arrow at the terminal point:

The same vector can be placed anywhere on the coordinate plane and still be the same vector -- the only two bits of information a vector represents are the magnitude and the direction. The magnitude is simply the length of the vector, and the direction is the angle at which it points. Since neither of these specify a starting or ending location, the same vector can be placed anywhere. To illustrate, all of the line segments below can be defined as the vector with magnitude $4{\sqrt {2}}$ and angle 45 degrees:

It is customary, however, to place the vector with the initial point at the origin as indicated by the black vector. This is called the standard position.

Component Form

In standard practice, we don't express vectors by listing the length and the direction. We instead use component form, which lists the height (rise) and width (run) of the vectors. It is written as follows:

${\begin{pmatrix}{\text{run}}\\{\text{rise}}\end{pmatrix}}$

Other ways of denoting a vector in component form include:

$\mathbf {(u_{x},u_{y})}$ and $\mathbf {\left\langle u_{x},u_{y}\right\rangle }$

From the diagram we can now see the benefits of the standard position: the two numbers for the terminal point's coordinates are the same numbers for the vector's rise and run. Note that we named this vector $\mathbf {u}$ . Just as you can assign numbers to variables in algebra (usually $x,y,z$), you can assign vectors to variables in calculus. The letters $u,v,w$ are usually used, and either boldface or an arrow over the letter is used to identify it as a vector.

When expressing a vector in component form, it is no longer obvious what the magnitude and direction are. Therefore, we have to perform some calculations to find the magnitude and direction.

Magnitude

$|\mathbf {u} |={\sqrt {u_{x}^{2}+u_{y}^{2}}}$

where $u_{x}$ is the width, or run, of the vector; $u_{y}$ is the height, or rise, of the vector. You should recognize this formula as the Pythagorean theorem. It is -- the magnitude is the distance between the initial point and the terminal point.

The magnitude of a vector can also be called the norm.

Direction

$\tan(\theta )={\frac {u_{y}}{u_{x}}}$

where $\theta$ is the counter-clockwise angle made by the vector with the positive $x$-axis. This formula is simply the tangent formula for right triangles.

Vector Operations

For these definitions, assume:

$\mathbf {u} ={\begin{pmatrix}u_{x}\\u_{y}\end{pmatrix}},\mathbf {v} ={\begin{pmatrix}v_{x}\\v_{y}\end{pmatrix}}$

Vector Addition

Vector Addition is often called tip-to-tail addition, because this makes it easier to remember.

The sum of the vectors you are adding is called the resultant vector, and is the vector drawn from the initial point (tip) of the first vector to the terminal point (tail) of the second vector. Although they look like the arrows, the pointy bit is the tail, not the tip. (Imagine you were walking the direction the vector was pointing... you would start at the flat end (tip) and walk toward the pointy end.)

It looks like this:

(Notice, the black lined vector is the sum of the two dotted line vectors!)

Numerically:

${\binom {4}{6}}+{\binom {1}{-3}}={\binom {5}{3}}$

Or more generally:

$\mathbf {u} +\mathbf {v} ={\begin{pmatrix}u_{x}+v_{x}\\u_{y}+v_{y}\end{pmatrix}}$

Scalar Multiplication

Graphically, multiplying a vector by a scalar changes only the magnitude of the vector by that same scalar. That is, multiplying a vector by 2 will "stretch" the vector to twice its original magnitude, keeping the direction the same.

$2\cdot {\binom {3}{3}}={\binom {6}{6}}$

Numerically, you calculate the resultant vector with this formula:

$c\mathbf {u} ={\begin{pmatrix}cu_{x}\\cu_{y}\end{pmatrix}}$ , where $c$ is a constant scalar.

As previously stated, the magnitude is changed by the same constant:

$|c\mathbf {u} |=|c||\mathbf {u} |$

Since multiplying a vector by a constant results in a vector in the same direction, we can reason that two vectors are parallel if one is a constant multiple of the other -- that is, that $\mathbf {u} ||\mathbf {v}$ if $\mathbf {u} =c\mathbf {v}$ for some constant $c$ .

We can also divide by a non-zero scalar by instead multiplying by the reciprocal, as with dividing regular numbers:

${\frac {\mathbf {u} }{c}}={\frac {1}{c}}\mathbf {u} ,c\neq 0$

Linear Functions

A depiction of the property of linearity with respect to a function applied to 2D vectors. In this figure, it is apparent that $L(\mathbf {u} +\mathbf {v} )=L(\mathbf {u} )+L(\mathbf {v} )$.

Given a function $L$ that accepts a vector as input and returns a vector or scalar as the output, function $L$ is considered to be "linear" if the following holds:

1. For any vectors $\mathbf {u}$ and $\mathbf {v}$, it is the case that $L(\mathbf {u} +\mathbf {v} )=L(\mathbf {u} )+L(\mathbf {v} )$.
2. For any vector $\mathbf {u}$ and scalar $c$, it is the case that $L(c\mathbf {u} )=cL(\mathbf {u} )$.

More generally, when given a function $M$ that has multiple vector valued parameters $\mathbf {u} _{1},\mathbf {u} _{2},...,\mathbf {u} _{n}$, function $M$ is a "multi-linear" function if $M$ is linear with respect to each parameter while holding all other parameters constant:

For each $i=1,2,...,n$ and vectors $\mathbf {u} _{1},\mathbf {u} _{2},\dots ,\mathbf {u} _{n}$:

1. For any vector $\mathbf {v}$ from the same vector space as $\mathbf {u} _{i}$, it is the case that $M(\mathbf {u} _{1},...,\mathbf {u} _{i}+\mathbf {v} ,...,\mathbf {u} _{n})=M(\mathbf {u} _{1},...,\mathbf {u} _{i},...,\mathbf {u} _{n})+M(\mathbf {u} _{1},...,\mathbf {v} ,...,\mathbf {u} _{n})$
2. For any scalar $c$, it is the case that $M(\mathbf {u} _{1},...,c\mathbf {u} _{i},...,\mathbf {u} _{n})=cM(\mathbf {u} _{1},...,\mathbf {u} _{i},...,\mathbf {u} _{n})$

If $n=2$, then $M$ is "bilinear". Bilinear functions include the dot product and the cross product.

Dot Product

The dot product is a way of multiplying two vectors to produce a scalar value. Because it combines the components of two vectors to form a /scalar/, it is sometimes called a scalar product. If you were asked to take the 'dot product of two rectangular vectors' you would do the following:

$\mathbf {u} \cdot \mathbf {v} =u_{x}v_{x}+u_{y}v_{y}$

It is very important to note that the dot product of two vectors does not result in another vector, it gives you a scalar, just a numerical value.

Another common pitfall may arise if your vectors are not in rectangular ('cartesian') format. Sometimes, vectors are instead expressed in polar coordinates, where the first component is the vector's magnitude (length) and the second is the angle from the $x$-axis at which the vector should be oriented. Dot products cannot be performed using the conventional method on these sorts of vectors; vectors in polar format must be converted to their equivalent rectangular form before you can work with them using the formula given above. A common way to convert to rectangular coordinates is to imagine that the vector was projected horizontally and vertically to form a right triangle. You could then use properties of sin and cos to find the length of the two legs the right triangle. The horizontal length would then be the x-component of the rectangular expression of the vector and the vertical length would be the y-component. Remember that if the vector is pointing down or to the left, the corresponding components would have to be negative to indicate that.

With some rearrangement and trigonometric manipulation, we can see that the number that results from the dot product of two vectors is a surprising and useful identity:

$\mathbf {u} \cdot \mathbf {v} =|\mathbf {u} ||\mathbf {v} |\cos(\theta )$

where $\theta$ is the angle between the two vectors.

Calculating bond angles of a symmetrical tetrahedral molecule such as methane using a dot product

This provides a convenient way of finding the angle between two vectors:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\theta)=\frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{u}||\mathbf{v}|}}

Notice that the dot product is 'commutative', that is:

$\mathbf {u} \cdot \mathbf {v} =\mathbf {v} \cdot \mathbf {u}$

Also, the dot product of two vectors will be the length of the vector squared:

$\mathbf {u} \cdot \mathbf {u} =u_{x}u_{x}+u_{y}u_{y}=(u_{x})^{2}+(u_{y})^{2}$

and by the Pythagorean theorem,

$(u_{x})^{2}+(u_{y})^{2}=\|u\|^{2}$

The dot product can be visualized as the length of a projection of one vector on to the other. In other words, the dot product asks 'how much magnitude of this vector is going in the direction of that vector?'

Deriving the Dot Product

Start with the following definition for the dot product: $\mathbf {u} \cdot \mathbf {v} =|\mathbf {u} ||\mathbf {v} |\cos(\theta )$ where $\theta$ is the angle between $\mathbf {u}$ and $\mathbf {v}$.

The formula $\mathbf {u} \cdot \mathbf {v} =u_{x}v_{x}+u_{y}v_{y}$ can be derived from the above definition through various approaches:

The triangle $\Delta AOB$.

Approach #1

One of the more direct approaches is to use the law of cosines. Create a triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta AOB} with vertices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(u_x,u_y)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(0,0)} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(v_x,v_y)} . The displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{OA} = \mathbf{u}} , the displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{OB} = \mathbf{v}} , and the displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{AB} = \mathbf{v}-\mathbf{u}} . The angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \angle AOB = \theta} .

The lengths of the sides of the triangle are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |OA| = |\mathbf{u}|} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |OB| = |\mathbf{v}|} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |AB| = |\mathbf{v}-\mathbf{u}|} . Applying the law of cosines gives:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |AB|^2 = |OA|^2 + |OB|^2 - 2|OA||OB|\cos(\angle AOB)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iff |\mathbf{v}-\mathbf{u}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2|\mathbf{u}||\mathbf{v}|\cos(\theta)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iff (v_x-u_x)^2 + (v_y-u_y)^2 = (u_x^2+u_y^2) + (v_x^2+v_y^2) - 2|\mathbf{u}||\mathbf{v}|\cos(\theta)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iff - 2u_xv_x - 2u_yv_y = -2|\mathbf{u}||\mathbf{v}|\cos(\theta)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iff |\mathbf{u}||\mathbf{v}|\cos(\theta) = u_xv_x + u_yv_y}

Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{v} = u_xv_x + u_yv_y} .

Approach #2

A more intuitive derivation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{v} = u_xv_x + u_yv_y} uses the fact that the dot product is a bilinear operator. To establish that the dot product is a bilinear operator, the following must be established:

1. Holding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} constant, the dot product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u} \cdot \mathbf{v}} must be linear with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} .
2. Holding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} constant, the dot product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u} \cdot \mathbf{v}} must be linear with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} .

Since it is readily apparent from the definition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos(\theta)} that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{v} = \mathbf{v}\cdot\mathbf{u}} , linearity with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} implies linearity with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} . It is hence only necessary to establish that the dot product is linear with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} to establish bilinearity.

The orthogonal projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} onto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\mathbf{u})} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos(\theta) = |\mathbf{u}|\text{proj}(\mathbf{v}|\mathbf{u})} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{proj}(\mathbf{v}|\mathbf{u}) = |\mathbf{v}|\cos(\theta)} is the "orthogonal projection" of vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} onto a line Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\mathbf{u})} whose direction is the direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{proj}(\mathbf{v}|\mathbf{u})} is the component of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} that is parallel to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\mathbf{u})} . It is not hard to observe that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{proj}(\mathbf{v}|\mathbf{u})} is linear with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} is held constant. The dot product $\mathbf {u} \cdot \mathbf {v} =|\mathbf {u} |{\text{proj}}(\mathbf {v} |\mathbf {u} )$ is linear with respect to $\mathbf {v}$, and therefore the dot product is a bilinear operator.

The bilinearity of the dot product now enables the derivation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{v} = \begin{pmatrix} u_x \\ u_y \end{pmatrix} \cdot \begin{pmatrix} v_x \\ v_y \end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left(u_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + u_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) \cdot \left(v_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + v_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left(\left(u_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + u_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix}\right)v_x + \left(\left(u_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + u_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)v_y} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left(\begin{pmatrix} 1 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix}\right)u_xv_x + \left(\begin{pmatrix} 0 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix}\right)u_yv_x + \left(\begin{pmatrix} 1 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)u_xv_y + \left(\begin{pmatrix} 0 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)u_yv_y} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 1u_xv_x + 0u_yv_x + 0u_xv_y + 1u_yv_y} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = u_xv_x + u_yv_y}

Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{v} = u_xv_x + u_yv_y} .

Applications of Scalar Multiplication and Dot Product

Unit Vectors

A unit vector is a vector with a magnitude of 1. The unit vector of u is a vector in the same direction as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} , but with a magnitude of 1:

The process of finding the unit vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} is called normalization. As mentioned in scalar multiplication, multiplying a vector by constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} will result in the magnitude being multiplied by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} . We know how to calculate the magnitude of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} . We know that dividing a vector by a constant will divide the magnitude by that constant. Therefore, if that constant is the magnitude, dividing the vector by the magnitude will result in a unit vector in the same direction as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}}  :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{w}=\frac{\mathbf{u}}{|\mathbf{u}|}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{w}} is the unit vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}}

Standard Unit Vectors

A special case of Unit Vectors are the Standard Unit Vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{i},\mathbf{j}}  : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{i}} points one unit directly right in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} direction, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}} points one unit directly up in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} direction:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{i}=\binom10}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}=\binom01}

Using the scalar multiplication and vector addition rules, we can then express vectors in a different way:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{x}{y}=x\mathbf{i}+y\mathbf{j}}

If we work that equation out, it makes sense. Multiplying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{i}} will result in the vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{x}{0}} . Multiplying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}} will result in the vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom0y} . Adding these two together will give us our original vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{x}{y}} . Expressing vectors using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{i},\mathbf{j}} is called standard form.

Projection and Decomposition of Vectors

Sometimes it is necessary to decompose a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} into two components: one component parallel to a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} } , which we will call Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}_\parallel}  ; and one component perpendicular to it, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}_\perp} .

Since the length of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}_\parallel} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathbf{u}|\cdot\cos(\theta)} , it is straightforward to write down the formulas for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}_\perp} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}_\parallel}  :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u_\parallel}=|\mathbf{u}|*\frac{(\mathbf{u}\cdot\mathbf{v})}{(|\mathbf{u}||\mathbf{v}|)}*\frac{\mathbf{v}}{|\mathbf{v}|}=\frac{ \mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}_\perp=\mathbf{u}-\mathbf{u}_\parallel}

Length of a vector

The length of a vector is given by the dot product of a vector with itself, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=0\ rad} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}\cdot\mathbf{u}=|\mathbf{u}||\mathbf{u}|\cos(\theta)=|\mathbf{u}|^2}

Perpendicular vectors

If the angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta } between two vectors is 90 degrees or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{\pi}{2}} (if the two vectors are orthogonal to each other), that is the vectors are perpendicular, then the dot product is 0. This provides us with an easy way to find a perpendicular vector: if you have a vector $\mathbf {u} ={\begin{pmatrix}u_{x}\\u_{y}\end{pmatrix}}$ , a perpendicular vector can easily be found by either

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}=\begin{pmatrix}-u_y\\u_x\end{pmatrix}=-\begin{pmatrix}u_y\\-u_x\end{pmatrix}}

Polar coordinates

Polar coordinates are an alternative two-dimensional coordinate system, which is often useful when rotations are important. Instead of specifying the position along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} axes, we specify the distance from the origin, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , and the direction, an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} .

Looking at this diagram, we can see that the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} are related to those of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} by the equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix}x=r\cos(\theta)&r=\sqrt{x^2+y^2}\\y=r\sin(\theta)&\theta=\arctan\left(\frac{y}{x}\right)\end{matrix}}

Because tan^-1 is multivalued, care must be taken to select the right value.

Just as for Cartesian coordinates the unit vectors that point in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} directions are special, so in polar coordinates the unit vectors that point in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} directions are also special.

We will call these vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{r}}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\boldsymbol{\theta}}} , pronounced r-hat and theta-hat. Putting a circumflex over a vector this way is often used to mean the unit vector in that direction.

Again, on looking at the diagram we see,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \mathbf{i}=\hat{\mathbf{r}}\cos(\theta)-\hat{\boldsymbol{\theta}}\sin(\theta)&\hat{\mathbf{r}}=\frac{x}{r}\mathbf{i}+\frac{y}{r}\mathbf{j}\\ \mathbf{j}=\hat{\mathbf{r}}\sin(\theta)+\hat{\boldsymbol{\theta}}\cos(\theta)&\hat{\boldsymbol{\theta}}=-\frac{y}{r}\mathbf{i}+\frac{x}{r}\mathbf{j} \end{matrix}}

Licensing

Content obtained and/or adapted from:

• Vectors, WikiBooks: Calculus under a CC BY-SA license