Question

If the vector a is multiplied by a negative number (- n) such that b=-n a then a and b have​

Answer 1

When a vector is multiplied by a negative number, what happens to its magnitude and direction?

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 viof 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.