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

Nov 6, 2015

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

#### Explanation:

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

$| | - 2 v | | = | - 2 | | | v | | = 2 | | v | | = 2 \cdot 3 = 6$.

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, $2 v$ is long two times the leght of $v$, and $\frac{1}{3} v$ is long one third of the lenght of $v$.

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