Question

If `hat"p", hat"q"` and `hat"r"` are unit vectors `hat"p"+hat "r" = hat "q"`, find `hat"p".hat"q".`

Answer 1

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`