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What are the smallest and largest values?

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2 Answers

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.
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)).

Comments

v and w are vectors, not numbers. v*w indicates the dot product of these vectors and llvll and llwll their magnitudes.
good catch. I misread the symbols for magnitude as absolute value. I'll make the necessary changes. :)