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

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

#### 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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