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

1 Answer
Nov 6, 2015

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.