Lila: /* Resources */

2021-11-02T16:04:40Z

Resources

Lila: /* Resources */

2021-10-01T19:39:31Z

Resources

Lila at 19:38, 1 October 2021

2021-10-01T19:38:08Z

Vu.nguyen: Created page with "* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018"

2020-10-30T14:46:21Z

Created page with "* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018"

New page

* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018

← Older revision		Revision as of 16:04, 2 November 2021
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	* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018		* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018
	* [https://en.wikibooks.org/wiki/Calculus/Vectors Vectors], WikiBooks: Calculus		* [https://en.wikibooks.org/wiki/Calculus/Vectors Vectors], WikiBooks: Calculus
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Vectors Vectors, WikiBooks: Calculus] under a CC BY-SA license

← Older revision		Revision as of 19:39, 1 October 2021
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	==Resources==		==Resources==
	* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018		* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018
		+	* [https://en.wikibooks.org/wiki/Calculus/Vectors Vectors], WikiBooks: Calculus

← Older revision		Revision as of 19:38, 1 October 2021
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		+	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:
		+	<center><math>\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y</math></center>
		+
		+	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 <math>x</math>-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:
		+
		+	<center><math>\mathbf{u}\cdot\mathbf{v}=\|\mathbf{u}\|\|\mathbf{v}\|\cos(\theta)</math></center>
		+
		+	where <math>\theta</math> is the angle between the two vectors.
		+	[[File:Tetrahedral angle calculation.svg\|thumb\|230px\|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:
		+	<center>
		+	<math>\cos(\theta)=\frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}</math></center>
		+	Notice that the dot product is 'commutative', that is:
		+	<center><math>\mathbf{u}\cdot\mathbf{v}=\mathbf{v}\cdot\mathbf{u}</math>
		+	</center>
		+	Also, the dot product of two vectors will be the length of the vector squared:
		+	<center><math>\mathbf{u}\cdot\mathbf{u}=u_xu_x+u_yu_y=(u_x)^2+(u_y)^2</math></center>
		+	and by the Pythagorean theorem,
		+	<center><math>(u_x)^2+(u_y)^2=\\|u\\|^2</math></center>
		+
		+	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?'
		+
		+
		+	==Resources==
	* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018		* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018

The Dot Product - Revision history

Lila: /* Resources */

Lila: /* Resources */

Lila at 19:38, 1 October 2021

Vu.nguyen: Created page with "* [https://www.youtube.com/watch?v=0iNrGpwZwog The Vector Dot Product ] Video by Professor Dave Explains 2018"