Multiplying vectors by scalars can accomplish two things: changing the length and/or reversing the direction of the vector.

## Multiplying Vectors by Scalars to Change Length

First, we can change the magnitude or length of the vector. If you have five vectors that each have a length of one meter, you can write this as a multiplication.

\(\color{black}{5 \times 1 \; \mathrm{m} \; \hat{x} = 5 \; \mathrm{m} \; \hat{x}}\)

Here’s what that looks like.

If you multiply by a number larger than one, the vector will become larger. If you multiply by a number less than one, the vector will become shorter. Division is handled the same way.

## Multiplying Vectors by Scalars to Reverse Direction

If you multiply a vector by negative one, the length stays the same, but the direction is reversed.

If you multiply a vector by a negative number other than one, you reverse its direction and change its length.

You can think of this vector as pointing in the negative direction. Occasionally, you might see people write \(\color{black}{2 \mathrm{m} \; (-\hat{x})}\). Either way is correct. Use the version that is easiest to understand in the context of the problem.