Question

If `|vec"a" + vec"b"| = |vec"a" - vec"b"|`, then the vectors `vec"a"` and `vec"b"` are orthogonal.

Options

• True

• False

Answer 1

This statement is True.

Explanation:

Given that `|vec"a" + vec"b"| = |vec"a" - vec"b"|`

Squaring both sides, we get

`|vec"a" + vec"b"|^2 = |vec"a" - vec"b"|^2`

⇒ `|vec"a"|^2 + |vec"b"|^2 + 2vec"a" * vec"b" = |vec"a"|^2 + |vec"b"|^2 - 2vec"a" * vec"b"`

⇒ `2vec"a" * vec"b" = -2vec"a" * vec"b"`

⇒ `vec"a" * vec"b" = -vec"a" * vec"b"`

⇒ `2vec"a" * vec"b"` = 0

⇒ `vec"a" * vec"b"` = 0

Which implies that `vec"a"` and `vec"b"` are orthogonal.