Description
A detailed tutorial of the dot product. Step by step tutorial including several examples of the dot product of a vector for reference.
Overview
The dot product of two vectors always ends up being a scalar. In mathematical terms, this is
Description
A detailed tutorial on how to solve vectors using vector subtraction. Step by step tutorial including several examples of vector subtraction for reference.
Overview
Vector subtraction involves two vectors that do not have to be equal, and could have different magnitudes and directions. The vectors are referred to as a and b. The formula for vector subtraction is:
In general, vector subtraction is defined geomtrically instead of algebraically, so it is not used quite as often as vector addition is.
![<span style="font-size: x-small;">\mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta[</span>/latex]. In this case, theta is the measure of the angle between a and b. The definition of a dot product given geometrically is that a and b have a common starting point and that the length of a is multiplied by the component in b that points in the same direction as a. Algebraically, it can be said that [latex]<span style="font-size: x-small;">\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.</span>](http://s.wordpress.com/latex.php?latex=%3Cspan%20style%3D%22font-size%3A%20x-small%3B%22%3E%5Cmathbf%7Ba%7D%5Ccdot%5Cmathbf%7Bb%7D%3D%5Cleft%5C%7C%5Cmathbf%7Ba%7D%5Cright%5C%7C%5Cleft%5C%7C%5Cmathbf%7Bb%7D%5Cright%5C%7C%5Ccos%5Ctheta%5B%3C%2Fspan%3E%2Flatex%5D.%20In%20this%20case%2C%20theta%20is%20the%20measure%20of%20the%20angle%20between%20a%20and%20b.%20The%20definition%20of%20a%20dot%20product%20given%20geometrically%20is%20that%20a%20and%20b%20have%20a%20common%20starting%20point%20and%20that%20the%20length%20of%20a%20is%20multiplied%20by%20the%20component%20in%20b%20that%20points%20in%20the%20same%20direction%20as%20a.%20Algebraically%2C%20it%20can%20be%20said%20that%20%5Blatex%5D%3Cspan%20style%3D%22font-size%3A%20x-small%3B%22%3E%5Cmathbf%7Ba%7D%20%5Ccdot%20%5Cmathbf%7Bb%7D%20%3D%20a_1%20b_1%20%2B%20a_2%20b_2%20%2B%20a_3%20b_3.%3C%2Fspan%3E&bg=ffffff&fg=000000&s=0 "<span style=\"font-size: x-small;\">\mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta[</span>/latex]. In this case, theta is the measure of the angle between a and b. The definition of a dot product given geometrically is that a and b have a common starting point and that the length of a is multiplied by the component in b that points in the same direction as a. Algebraically, it can be said that [latex]<span style=\"font-size: x-small;\">\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.</span>")
