Question

If ||v|| = 3, what is ||-2v||?

Answer 1

The norm of `-2v` is `6`

Explanation:

Using formulas, you know that `||\lambda v|| = |lambda| ||v||`, so

`||-2v|| = |-2| ||v|| = 2||v|| = 2*3=6`.

Intuitively, the norm measures the leght of a vector, and multiplying a vector by a number means to stretch (or shrink) the vector according to the number. For example, `2v` is long two times the leght of `v`, and `1/3 v` is long one third of the lenght of `v`.

The sign of the number only affect the direction of the vector: `5v` is long five times the original lenght and has the same orientation as `v`, while `-9v` is long nine times the lenght of `v` and has the opposite orientation.