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:

\[P Q^2 = O P^2 + O Q^2 - 2OP \cdot OQ\cos\alpha\]

\[\Rightarrow \left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2 = {x_1}^2 + {y_1}^2 + {x_2}^2 + {y_2}^2 - 2OP \cdot OQ\cos\alpha\]

\[\Rightarrow {x_2}^2 + {x_1}^2 - 2 x_1 x_2 + {y_2}^2 + {y_1}^2 - 2 y_1 y_2 = {x_1}^2 + {y_1}^2 + {x_2}^2 + {y_2}^2 - 2OP \cdot OQ\cos\alpha\]

\[\Rightarrow - 2 x_1 x_2 - 2 y_1 y_2 = - 2OP \cdot OQ\cos\alpha\]

\[\Rightarrow OP \cdot OQ\cos\alpha = x_1 x_2 + y_1 y_2\]