Three-Dimensional Coordinate Systems

Three-Dimensional Coordinates and Vectors

Basic definition

Two-dimensional Cartesian coordinates as we've discussed so far can be easily extended to three-dimensions by adding one more value: $z$ . If the standard $(x,y)$ coordinate axes are drawn on a sheet of paper, the $z$-axis would extend upwards off of the paper.

Similar to the two coordinate axes in two-dimensional coordinates, there are three coordinate planes in space. These are the $xy$-plane, the $yz$-plane, and the $xz$-plane. Each plane is the "sheet of paper" that contains both axes the name mentions. For instance, the $yz$-plane contains both the $y$ and $z$ axes and is perpendicular to the $x$-axis.

Therefore, vectors can be extended to three dimensions by simply adding the $z$ value.

$\mathbf {u} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}$

To facilitate standard form notation, we add another standard unit vector:

$\mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}$

Again, both forms (component and standard) are equivalent.

${\begin{pmatrix}1\\2\\3\end{pmatrix}}=1\mathbf {i} +2\mathbf {j} +3\mathbf {k}$

Magnitude: Magnitude in three dimensions is the same as in two dimensions, with the addition of a $z$ term in the radicand.

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

Three dimensions

The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate systems, both of which include two-dimensional or planar polar coordinates as a subset. In essence, the cylindrical coordinate system extends polar coordinates by adding an additional distance coordinate, while the spherical system instead adds an additional angular coordinate.

Cylindrical coordinates

a point plotted with cylindrical coordinates

The cylindrical coordinate system is a coordinate system that essentially extends the two-dimensional polar coordinate system by adding a third coordinate measuring the height of a point above the plane, similar to the way in which the Cartesian coordinate system is extended into three dimensions. The third coordinate is usually denoted $h$ , making the three cylindrical coordinates $(r,\theta ,h)$ .

The three cylindrical coordinates can be converted to Cartesian coordinates by

${\begin{aligned}x&=r\cos(\theta )\\y&=r\sin(\theta )\\z&=h\end{aligned}}$

Spherical coordinates

A point plotted using spherical coordinates

Polar coordinates can also be extended into three dimensions using the coordinates $(\rho ,\phi ,\theta )$ , where $\rho$ is the distance from the origin, $\phi$ is the angle from the $z$-axis (called the colatitude or zenith and measured from 0 to 180°) and $\theta$ is the angle from the $x$-axis (as in the polar coordinates). This coordinate system, called the spherical coordinate system, is similar to the latitude and longitude system used for Earth, with the origin in the centre of Earth, the latitude $\delta$ being the complement of $\phi$ , determined by $\delta =90^{\circ }-\phi$ , and the longitude $l$ being measured by $l=\theta -180^{\circ }$ .

The three spherical coordinates are converted to Cartesian coordinates by

${\begin{aligned}x&=\rho \sin(\phi )\cos(\theta )\\y&=\rho \sin(\phi )\sin(\theta )\\z&=\rho \cos(\phi )\end{aligned}}$

$\rho ={\sqrt {x^{2}+y^{2}+z^{2}}}$

$\theta =\arctan \left({\frac {y}{x}}\right)$

$\phi =\arccos \left({\frac {z}{\rho }}\right)$

Cross Product

The cross product of two vectors is a determinant:

$\mathbf {u} \times \mathbf {v} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\u_{x}&u_{y}&u_{z}\\v_{x}&v_{y}&v_{z}\end{vmatrix}}$

and is also a pseudovector.

The cross product of two vectors is orthogonal to both vectors. The magnitude of the cross product is the product of the magnitude of the vectors and the sin of the angle between them.

$|\mathbf {u} \times \mathbf {v} |=|\mathbf {u} ||\mathbf {v} |\sin(\theta )$

This magnitude is the area of the parallelogram defined by the two vectors.

The cross product is linear and anticommutative. For any numbers $a,b$ ,

$\mathbf {u} \times \left(a\mathbf {v} +b\mathbf {w} \right)=a\mathbf {u} \times \mathbf {v} +b\mathbf {u} \times \mathbf {w} \qquad \mathbf {u} \times \mathbf {v} =-\mathbf {v} \times \mathbf {u}$

If both vectors point in the same direction, their cross product is 0.

Deriving the Cross Product

A depiction of the cross product of vectors $\mathbf {u}$ and $\mathbf {v}$.

Start with the following definition for the cross 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}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}}} . 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 \hat{\mathbf{n}}} is a unit length vector that is perpendicular to both 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}} 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{v}} and oriented according to the "right hand rule". 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} is the counterclockwise angle from 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}} 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}} within the plane that is orthogonal 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 \hat{\mathbf{n}}} .

The formula 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}\times\mathbf{v} = (u_yv_z-u_zv_y)\mathbf{i} + (u_zv_x-u_xv_z)\mathbf{j} + (u_xv_y-u_yv_x)\mathbf{k}} can be derived by exploiting the bilinearity of the cross product. To establish the cross product as 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 cross product $\mathbf {u} \times \mathbf {v}$ must be linear with respect to $\mathbf {v}$.
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 cross 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}\times\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}} .

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}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}}} , the anticommutative property of the cross product can be inferred: 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}\times\mathbf{v} = -\mathbf{v}\times\mathbf{u}} (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 \hat{\mathbf{n}}} reverses direction when 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}} 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{v}} are swapped). This means that 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 cross 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 perpendicular 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}} relative to the 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})} .

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}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}} = |\mathbf{u}|\textbf{rotate}(\textbf{perp}(\mathbf{v}|\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 \textbf{perp}(\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 perpendicular to 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 \textbf{perp}(\mathbf{v}|\mathbf{u}) = (|\mathbf{v}|\sin\theta)\hat{\mathbf{m}}} where 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 \hat{\mathbf{m}}} is a unit length vector 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 \textbf{perp}(\mathbf{v}|\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 \textbf{rotate}(\mathbf{w}|\mathbf{u})} is a 90 degree counterclockwise rotation 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{w}} around 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})} . Note 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 \textbf{rotate}(\hat{\mathbf{m}}|\mathbf{u}) = \hat{\mathbf{n}}} .

It can easily be observed 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 \textbf{perp}(\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}} with 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}} held constant, and 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 \textbf{rotate}(\mathbf{w}|\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{w}} with 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}} held constant. The cross 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}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}} = |\mathbf{u}|\textbf{rotate}(\textbf{perp}(\mathbf{v}|\mathbf{u})|\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}} , and therefore the cross product is a bilinear operator.

The bilinearity of the cross 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}\times\mathbf{v} = (u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times(v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k})} 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 = v_x((u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times\mathbf{i}) + v_y((u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times\mathbf{j}) + v_z((u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times\mathbf{k})} 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(\mathbf{i}\times\mathbf{i}) + u_yv_x(\mathbf{j}\times\mathbf{i}) + u_zv_x(\mathbf{k}\times\mathbf{i}) + u_xv_y(\mathbf{i}\times\mathbf{j}) + u_yv_y(\mathbf{j}\times\mathbf{j}) + u_zv_y(\mathbf{k}\times\mathbf{j}) + u_xv_z(\mathbf{i}\times\mathbf{k}) + u_yv_z(\mathbf{j}\times\mathbf{k}) + u_zv_z(\mathbf{k}\times\mathbf{k})} 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_y\mathbf{0} + u_yv_x(-\mathbf{k}) + u_zv_x\mathbf{j} + u_xv_y\mathbf{k} + u_yv_y\mathbf{0} + u_zv_y(-\mathbf{i}) + u_xv_z(-\mathbf{j}) + u_yv_z\mathbf{i} + u_zv_z\mathbf{0}} 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_yv_z-u_zv_y)\mathbf{i} + (u_zv_x-u_xv_z)\mathbf{j} + (u_xv_y-u_yv_x)\mathbf{k}}

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}\times\mathbf{v} = (u_yv_z-u_zv_y)\mathbf{i} + (u_zv_x-u_xv_z)\mathbf{j} + (u_xv_y-u_yv_x)\mathbf{k}} .

Triple Products

If we have three vectors we can combine them in two ways, a triple scalar 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}\times\mathbf{w})}

and a triple vector 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}\times(\mathbf{v}\times\mathbf{w})}

The triple scalar product is a determinant

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}\times\mathbf{w})=\begin{vmatrix}u_x&u_y&u_z\\v_x&v_y&v_z\\w_x&w_y&w_z\end{vmatrix}}

If the three vectors are listed clockwise, looking from the origin, the sign of this product is positive. If they are listed anticlockwise the sign is negative.

The order of the cross and dot products doesn't matter.

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}\times\mathbf{w})=(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}}

Either way, the absolute value of this product is the volume of the parallelepiped defined by the three 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{u},\mathbf{v},\mathbf{w}}

The triple vector product can be simplified

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}\times(\mathbf{v}\times\mathbf{w})=(\mathbf{u}\cdot\mathbf{w})\mathbf{v}-(\mathbf{u}\cdot\mathbf{v})\mathbf{w}}

This form is easier to do calculations with.

The triple vector product is not associative.

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}\times(\mathbf{v}\times\mathbf{w})\ne(\mathbf{u}\times\mathbf{v})\times\mathbf{w}}

There are special cases where the two sides are equal, but in general the brackets matter. They must not be omitted.

Three-Dimensional Lines and Planes

We will use 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 r} to denote the position of a point.

The multiples of 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 a} all lie on a line through the origin. Adding a constant 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 b} will shift the line, but leave it straight, so the equation of a line is,

$\mathbf {r} =s\mathbf {a} +\mathbf {b}$

This is a parametric equation. The position is specified in terms of the parameter 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 s} .

Any linear combination of 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 \mathbf{a},\mathbf{b}} lies on a single plane through the origin, provided the two vectors are not colinear. We can shift this plane by a constant vector again and write

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{r}=s\mathbf{a}+t\mathbf{b}+\mathbf{c}}

If we choose 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{a},\mathbf{b}} to be orthonormal vectors in the plane (i.e. unit vectors at right angles) then 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 s,t} are Cartesian coordinates for points in the plane.

These parametric equations can be extended to higher dimensions.

Instead of giving parametric equations for the line and plane, we could use constraints. E.g., for any 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 xy} plane 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 z=0}

For a plane through the origin, the single vector normal to the plane, 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 n} , is at right angle with every vector in the plane, by definition, so

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{r}\cdot\mathbf{n}=0}

is a plane through the origin, normal 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 n} .

For planes not through the origin we get

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{r}-\mathbf{a})\cdot\mathbf{n}=0\qquad\mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n}}

A line lies on the intersection of two planes, so it must obey the constraint for both planes, i.e.

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{r}\cdot\mathbf{n}=a\qquad\mathbf{r}\cdot\mathbf{m}=b}

These constraint equations con also be extended to higher dimensions.

Resources

• Plotting Points In a Three Dimensional Coordinate System Video by The Organic Chemistry Tutor 2017

Licensing

Content obtained and/or adapted from:

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