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Question
If `hat"p", hat"q"` and `hat"r"` are unit vectors `hat"p"+hat "r" = hat "q"`, find `hat"p".hat"q".`
Solution
Let the triangle be denoted by ABC, where `"AB" = hat"p"`, `"AC" = hat"q"` and `"BC" = hat"r"`
∵ `hat"p", hat"q", hat"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 A = `(hat"p".hat"q")/(|hat"p"|.|hat"q"|)`
∴ cos 60° = `(hat"p".hat"q")/(1 xx 1)`
∴ `1/2 = hat"p".hat"q"`
∴ `hat"p".hat"q" = 1/2`
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