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If pqp¯,q¯ and rr¯ are unit vectors, find prp¯.r¯. - Mathematics and Statistics

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Sum

If `bar"p", bar"q"` and `bar"r"` are unit vectors, find `bar"p".bar"r".`

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Solution

Let the triangle be denoted by ABC, where `bar"AB" = bar"p"`, `bar"AC" = bar"q"` and `bar"BC" = bar"r"`

∵ `bar"p", bar"q", bar"r"` are unit vectors.

∴ l(AB) = l(BC) = l(CA) = 1

∴ the triangle is equilateral

∴ ∠A = ∠B = ∠C = 60°

Using the formula for angle between two vectors,

cos B = `(bar"p".bar"r")/(|bar"p"|.|bar"r"|)`

we get `bar"p".bar"r" = 1/2`

∴ cos 60° = `(bar"p".bar"r")/(1 xx 1)`

∴ `1/2 = bar"p".bar"r"`

∴ `bar"p".bar"r" = 1/2`

Concept: Vector Product of Vectors (Cross)
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