Question

when a vector is multiplied by a negative number its direction changes at an angle of....

Answer 1

Explanation:

One way to express the magnitude (length) of a vector  V⃗ ∈Rn  is by extending Pythagorean’s Theorem to  n -dimensions. Hence we can write the length of  V⃗   as  ||V⃗ ||=v21+v22+⋯+v2n−1+v2n−−−−−−−−−−−−−−−−−−−−√ , where each  vi  is a component of  V⃗ .  

Observe that if we scale  V⃗   by a factor of  c∈R , then each component  vi of  V⃗   becomes  cvi.  So the length of  ||cV⃗ ||=  

(cv1)2+(cv2)2+⋯+(cvn−1)2+(cvn)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=  

c2v21+c2v22+⋯+c2v2n−1+c2v2n−−−−−−−−−−−−−−−−−−−−−−−−−−√=  

c2−−√v21+v22+⋯+v2n−1+v2n−−−−−−−−−−−−−−−−−−−−√=  

|c|v21+v22+⋯+v2n−1+v2n−−−−−−−−−−−−−−−−−−−−√=|c| ||V⃗ || .

In your case,  c=−n , so for any vector  V⃗ , the length of  −nV⃗   is given by  ||−nV⃗ ||=|−n|||V⃗ ||=n||V⃗ || . Ultimately, the sign of a scalar will have no effect on the magnitude, since length is always a non-negative measurement.

On the other hand, scaling  V⃗   by any  c<0  reverses its direction, and scaling  V⃗   by  c=0  destroys it.

Hope it helps!!!