If the line segment joining the points p(x1,y1) and Q(x2,y2)subtends and angle α at the origin O,prove that:OP.OQ cos α=x1 x2+y1 y2.
It is given that O is the origin,Then,OQ2=x22+y22OP2=x21+y21and PQ2=(x2−x1)2+(y2−y1)2Using cosine formula in ΔOPQ,we havePQ2=OP2+OQ2−2(OP)(OQ)cos α⇒(x2−x1)2+(y2−y1)2=x22+y22+x21+y21−2(OP),(OQ) cosα⇒x22+x21−2x2x1+y22+y21−2y2y1=x22+y22+x21+y21−2OP.OQ cosα⇒−2x1x2−2y1y2=−2OP.OQ cosα⇒x1x2+y1y2=OP.OQ cosα⇒OP.OQ cosα=x1x2+y1y2 Hence,proved.