The resultant of two vectors `vec(P)` and `vec(Q)` is `vec(R )`. If the magnitude of `vec(Q)` is doubled, the new resultant vector becomes perpendicular to `vec(P)`. Then, the magnitude of `vec(R )` is equal to

Options

A. ` P+Q`
B. `Q`
C. `P`
D. ` (P+Q)/2`

Answer

B

Explanation

.
` tan beta = (Q sin theta)/( P + Q cos theta) and `
` R^2= P^2 + Q P Q cos theta` ….(i)
When ` (Q) is doubled, resultant ` vec R_1` is perpendicular to ` vec P` . Fig. 2 (CF). 51 (b). The
` R_1^2 = P^2 + 4 Q^2 + 4 PQ cos thea` ....(ii)
From right angled triangle ` BAD_1`, we have
` R-1^2 = ( 2 Q)^2 - P^2 = 4 Q^2 -P^2`
Putting this value in (ii), we get
` Q^@ + 2 - P^2 =P^2 + 4 Q^2 + 4 PQ cos theta` ltbRgt or ` P^2 + 2 PQ cos theta =0`
Putting this value in (ui), we get
` R^2 = (P^2 +2 PQ cos theta =0`
Putting this value in (i), we get .
` R^1 = (P^2 + 2 PQ cos theta) + Q^2 =0 + Q^2 =Q^2 ` or ` R=Q`.