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Question

If be the origin, and if the coordinates of any two points P1 and P2 respectively (x1,y2) and(x2,y2), prove that
OP1OP2cosP1OP2=x1x2+y1y2

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Solution

cosP1OP2=(OP1)2+(OP2)2(P1P2)22(OP1)(OP2)
We have points P1(x1,y1), P2(x2,y2) and O(0,0)
Then,
OP1=x21+y21
OP2=x22+y22
P1P2=(x1x2)2+(y1y2)2
Substituting the above results in the given equation, we get
OP1.OP2.cos(P1OP2)=(x21+y21)+(x22+y22)[(x1x2)2+(y1y2)2]2=x1x2+y1y2
Hence proved

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