Adding Vectors

Like numbers, vectors can be added together, by lining them up in a chain, and then drawing a vector from the beginning of the chain to the end. It doesn't matter what order you line up the vectors, as long as you place them "tip-to-tail".

Multiplying a vector by a positive number changes the magnitude of the vector while leaving its direction unchanged: \(\vec A\), \(3\vec A\), and \(0.1\vec A\) all point in the same direction.

Multiplying a vector by a negative number reverses its direction. We use this to define vector subtraction:

$$\vec A-\vec B=\vec A+(-\vec B)$$
So for example, as we see in the figure, if \(\vec b+\vec a=\vec c,\) we can subtract the vector \(\vec a\) from both sides, giving us \(\vec b=\vec c-\vec a\), or \(\vec b=\vec c+(-\vec a).\)

In the following interactive demonstration, try to find combinations of \(\vec A\) and \(\vec B\) which add up to the black vector shown, in order to get a feel for how vector addition works.

Interactive 1.2.1