What are the smallest and largest values?

Question

If llvll=5 and llwll=3, what are the smallest and largest values of llv-wll? What are the smallest and largest values of v*w?

Answer 1

Assuming v and w are vectors with any direction but with definite magnitudes of 5 and 3 respectively, the largest value for the magnitude of vector v-w will occur when v and w are oriented parallel to each other pointed in the same direction giving a magnitude of v-w = mag(v) + mag(w). The smallest will occur when v and w are oriented parallel to each other but in opposite directions resulting in a mag for v-w equal to abs(mag(v) - mag(w)).

Answer 2

If llvll=5 and llwll=3,

a) the smallest value of llv-wll is 5-3=2, which happens when v and w are parallel.

b) the largest value of llv-wll is 5+3=8, which happens when v and w are anti-parallel.

c) the smallest value of v*w is 5*3*(-1)=-15, which happens when v and w are anti-parallel (cos 180°=-1).

d) the largest value of v*w is 5*3*(1)=15, which happens when v and w are parallel (cos 0°= 1).

 

Remember that v*w= llvll llwll cosθ, and abs(cosθ)≤1.