Question

Prove that two vectors are perpendicular iff |a+b|^2=|a|^2+|b|^2​

Answer 1

Answer:

(Aah, just solved this question from NCERT Miscellaneous Exercise :p)

To begin with the proof,

(a+b)^2= a·a + 2a·b + b·b

If (a+b)^2 = a^2 + b^2, then

⇒ 2a·b = 0

The dot product of two non-zero vectors is zero only when they are perpendicular. Thus, for non-zero vectors a and b,

(a+b)^2 = a^2 + b^2 ⇒ a·b = 0 ⇒ a⊥b

Hence proved.