Question

If the line segment joining the points P (x₁, y₁) and Q (x₂, y₂) subtends an angle α at the origin O, prove that
OP · OQ cos α = x₁ x₂ + y₁, y₂

Answer 1

From the figure,
$O{P}^2={x_1}^2+{y_1}^2$
$O{Q}^2={x_2}^2+{y_2}^2$
$P{Q}^2=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$
Using cosine formula in $∆OPQ$, we get:
$P{Q}^2=O{P}^2+O{Q}^2-2OP·OQ\cos \alpha$
$\Rightarrow {\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2={x_1}^2+{y_1}^2+{x_2}^2+{y_2}^2-2OP·OQ\cos \alpha$
$\Rightarrow {x_2}^2+{x_1}^2-2x_1{x}_2+{y_2}^2+{y_1}^2-2y_1{y}_2={x_1}^2+{y_1}^2+{x_2}^2+{y_2}^2-2OP·OQ\cos \alpha$
$\Rightarrow -2x_1{x}_2-2y_1{y}_2=-2OP·OQ\cos \alpha$
$\Rightarrow OP·OQ\cos \alpha =x_1{x}_2+y_1{y}_2$