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If O is the origin and if the coordina...

If `O` is the origin and if the coordinates of any two points `Q_1 \ a n d \ Q_2` are `(x_1,y_1) \ a n d \ (x_2,y_2),` respectively, prove that `O Q_1.O Q_2cos/_Q_1O Q_2=x_1x_2+y_1y_2`.

Text Solution

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In `DeltaOQ_1Q_2`, using cosing rule, we get
`(Q-1Q_2)^2=(OQ_1)^2+(OQ_2)^2-OQ_1.OQ_2.cos(angleQ_1OQ_2)`
`therefore(x_2-x_1)^2+(y_2-y_1)^2`
`=(x_(1)^2+y_1^2)+(x_2^2+y_2^2)-2OQ_1.OQ_2.cos(angleQ_1OQ_2)`
`rArrOQ_1.OQ_2cos(angleQ_1OQ_2)=x_1x_2+y_1y_2`.
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