Two vector ` vec P` and ` vec Q` act at a point and have a resultant ` R_1. If `vec Q` is replaced by the vevtor ` ( vec (R_1^2-P^2)/Q` show tht the reaultant is still of magnitude ` vec R_1`.

Explanation

Refer Fing. 2 (c ) . 97.,let ` theta ` be the angle between
` vec P ` and ` vec Q`. Then` vec R-1 ^2 =P^2 + Q^2 + 2 PQ cos theta `
or ` 2 PQ cos theta = R_1^2 -P-Q^2` ..(i)
When the vector ` vec Q` is replaced by the vector
` (( vec (R_1^2 - P^2)/Q)` in a direction opposite to that of
` vec Q`, then the angle between the vectors is ltbRgt (180^2- theta)`. If ` vec R_2` is the magnitude of the new resultant. then
.
` R_2^2 = P^2 + ((R_1^2 - P^2)/Q()^2 + 2 P ( (R_1^2 - P^2)/Q) cos ( 180^@ -theta)`
` = 1/Q^2 [ P^2 Q^2 + R_1^4 - 2 R_1^2 P^2 + P^4`
` - 2 P Q (R-1^2 - P^2) cos theta ]`
` = 1/Q^2 [ P^2 Q^2 +R_1^4 - 2 R_1^2 P^2 +P^4`
` ( R_1^2-P^2) (R_1^2-P^2-Q^2)] [From (i)]`
`= 1 /Q^2 [P^2 Q^2+R_1^2-R_1^2 P^2+P^4`
`- R_!^4 + 2 P^4 R_1^2 -P^4 -P^4 Q^2 +R_1^2Q^2]`
`= (R_1^2 Q^2)/Q^@ = R_1^2`
:. ` R_2 =R_1`. Thus the magniude of resultant in the second case is also ` R_1`.