1.

If O be the origin and `A(x_(1), y_(1)), B(x_(2), y_(2))` are two points, then what is `(OA) (OB) cos angle AOB` ?A. `x_(1)^(2)+x_(2)^(2)`B. `y_(1)^(2)+y_(2)^(2)`C. `x_(1)x_(2)+y_(1)y_(2)`D. `x_(1)y_(1)+x_(2)y_(2)`

Answer» Correct Answer - C
Let `O(0, 0), A(x_(1), y_(1))` and `B(x_(2), y_(2))` be three points
`OA=sqrt(x_(1)^(2)+y_(1)^(2)), OB=sqrt(x_(2)^(2)+y_(2)^(2))`
`AB=sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2))`
In `Delta AOB`,
`s`
`cos angle AOB=(OA^(2)+OB^(2)-AB^(2))/(2.OA.OB)`
`implies OA.OB cos angle AOB=(OA^(2)+OB^(2)-AB^(2))/2`
`=(x_(1)^(2)+y_(1)^(2)+x_(2)^(2)+y_(2)^(2)-{(x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)})/2`
`implies OA.OB. cos angle AOB`
`=(x_(1)^(2)+y_(1)^(2)+x_(2)^(2)+y_(2)^(2)-{x_(2)^(2)+x_(1)^(2)-2x_(1)x_(2)+y_(2)^(2)+y_(1)^(2)-2y_(1)y_(2)})/2`
`=(2(x_(1)x_(2)+y_(1)y_(2)))/2=x_(1)x_(2)+y_(1)y_(2)`


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