In an exam, i was asked to prove that the bisectors of two vectors, v + w and v - w, are perpendicular given that v and w have the same magnitude and different from zero. I did this:
||v||=||w||
b1=v||w|| + w||v|| = v||v||+ w||v||=(v + w)||v||
b2=v||w|| - w||v|| = v||v|| - w||v|| = (v - w)||v||
b1 • b2 $\overset !=$ (v + w) • (v - w)= v•v - v•w + v•w - w•w =v•v - w•w= ||v||² - ||w||² = ||v||² - ||v||² = 0
I dont understand why the highlighted equal sign isnt correct. I only got that i needed to prove it and lost a couple of marks
You have defined $b_1 = (v+w)\|v\|$ and $b_2 = (v-w)\|v\|$ in the first two lines. Thus, we have $$ b_1 \cdot b_2 = [\|v\|(v+w)] \cdot [\|v\|(v-w)] $$ which will only be equal to $(v+w)\cdot(v-w)$ if $\|v\| = 1$.
Incidentally, I suspect that if you had simply written
(v + w) • (v - w)= v•v - v•w + v•w - w•w =v•v - w•w= ||v||² - ||w||² = ||v||² - ||v||² = 0
and wrote "two vectors are orthogonal if their dot product is zero", then you would have received full credit.