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Question
If `|vec"a" + vec"b"| = |vec"a" - vec"b"|`, then the vectors `vec"a"` and `vec"b"` are orthogonal.
Options
True
False
Solution
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.
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