Prove that , if and only if are perpendicular, given .

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Solution

It is given that a → ≠ 0 → and b → ≠ 0 → .

Calculate the dot product of a → + b → with itself.

( a → + b → )⋅( a → + b → )= a → ⋅ a → + a → ⋅ b → + b → ⋅ a → + b → ⋅ b → = a → ⋅ a → + a → ⋅ b → + a → ⋅ b → + b → ⋅ b → = a → ⋅ a → +2 a → ⋅ b → + b → ⋅ b → = | a → | 2 +2 a → ⋅ b → + | b → | 2

It is known that if a → ⋅ b → =0, then a → and b → are perpendicular.

( a → + b → )⋅( a → + b → )= | a → | 2 +2⋅0+ | b → | 2 = | a → | 2 + | b → | 2

Hence, it is proved that ( a → + b → )⋅( a → + b → )= | a → | 2 + | b → | 2 , if and only if a → and b → are perpendicular.

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