If the line segment joining the points `P(x_1, y_1)a n d\ Q(x_2, y_2)` subtends an angle `alpha` at the origin O, prove that : `O PdotO Q\ cosalpha=x_1x_2+y_1y_2dot`

Explanation

We have given the line segment joining the points `P(x_1, y_1)a n d\ Q(x_2, y_2)`subtends an angle `alpha` at the origin O

Now let `a^2=PQ=(x_2-x_1)^2+(y_2-y_1)^2,
b^2=OQ=(x_2)^2+(y_2)^2,
c^2=OP=(x_1)^2+(y_1)^2`,

Then by cosine formula we have,

`cosalpha=(b^2+c^2-a^2)/(bc)`

`cosalpha=(((x_2)^2+(y_2)^2)+((x_1)^2+(y_1)^2)-((x_2-x_1)^2+(y_2-y_1)^2))/((sqrt((x_2)^2+(y_2)^2)sqrt((x_1)^2+(y_1)^2)))`

`cosalpha=(2(x_1x_2+y_1y_2))/(2OP*OQ)`

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