Question

Show that if u+v and u-v are orthognal, then the vectors u and v must have the same length.

Answer 1

Answer with Step-by-step explanation:

We are given that

u+ v and u-v are orthogonal

We have to prove that u and v must have the same length.

When two vector a and b are orthogonal then

$a\cdot b=0$

By using the property

$(u+v)\cdot (u-v)=0$

We know that

$(a+b)\cdot (a-b)=\mid a\mid^2-\mid b\mid^2$

$\mid u\mid ^2-\mid v\mid^2=0$

$\mid u\mid^2=\mid v\mid^2$

Magnitude is always positive

When power of base on both sides are equal then base will be equal

Therefore,

$\mid u\mid=\mid v\mid$

Hence, the length of vectors u and v must have the same length.