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Vector addition is a simple yet important operation. To find the sum of two vectors, called a net vector, just add the respective components. In other words, add the first components of each vector, then add the second components, and so forth, and the results are the components of the vector sum.

In general for adding three-dimensional vectors,

Graphically, vector addition can be shown by drawing the vectors tip-to-tail (the tip is the end of the arrowhead and the tail is from where it starts). From the origin, draw the first vector, then from the end of it, draw the second, etc. After you’ve drawn them all, the net vector starts at the origin and ends where the last one drawn ends. This CDF demonstrates this by adding the blue, red, and green vectors (you can drag the tips of them around the pane) to result in the black vector. In addition to drawing the vectors tip-to-tail, the program draws the components in the same manner and ends up with the same result. Just like normal addition, vector addition is commutative (the order doesn’t matter). This is true for calculating the sum as well as drawing them.

Any two vectors can be added as long as they have the same number of dimensions.You also can’t add a scalar to a vector. You can subtract vectors in the same manner, however. If you wish to do so graphically, just align the vectors tip-to-tip instead, and the net vector will be from the tail of the first to the tail of the second. Like normal subtraction, this is not commutative.

Vector addition is commonly used with movement and force analysis. With several displacements, you can find a net displacement simply by adding up the individual vectors. If several forces are acting on an object, you can just find the sum of the vector forces and find the net force. You can then treat the object like the net force is the only one acting on it.

Alpha is handy for handling vector math. Vectors are notated with braces and it’s very simple to add them. Here’s an example:

{2,6}+{1,-2}

Or you can just use this widget, which was created using Wolfram Alpha. Check out the how to post to find out how to make one for yourself.