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If aa¯ and bb¯ are two vectors perpendicular to each other, prove that abab(a¯+b¯)=(a¯-b¯) - Mathematics and Statistics

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Sum

If `bar"a"` and `bar"b"` are two vectors perpendicular to each other, prove that `(bar"a" + bar"b")^2 = (bar"a" - bar"b")^2`

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Solution

`bar"a"` and `bar"b"` are perpendicular to each other.

∴ `bar"a".bar"b" = bar"b".bar"a" = 0`    ...(1)

LHS = `(bar"a" + bar"b")^2`

`= (bar"a" + bar"b").(bar"a" + bar"b")`

`= bar"a".(bar"a" + bar"b") + bar"b"(bar"a" + bar"b")`

`= bar"a".bar"a" + bar"a".bar"b" + bar"b".bar"a" + bar"b".bar"b"`

`= bar"a".bar"a" + 0 + 0 + bar"b".bar"b"`   ....[By (1)]

`= |bar"a"|^2 + |bar"b"|^2`

RHS = `(bar"a" - bar"b")^2`

`= (bar"a" - bar"b").(bar"a" - bar"b")`

`= bar"a".(bar"a" - bar"b") + bar"b"(bar"a" - bar"b")`

`= bar"a".bar"a" - bar"a".bar"b" - bar"b".bar"a" + bar"b".bar"b"`

`= bar"a".bar"a" + bar"b".bar"b"`     ...[By(1)]

`= |bar"a"|^2 + |bar"b"|^2`

∴ LHS = RHS

Hence, `(bar"a" + bar"b")^2 = (bar"a" - bar"b")^2`

Concept: Vector Product of Vectors (Cross)
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