# The resultant of two vectors P(arrowhead) and Q(arrowhead) has the magnitude of P(arrowhead). The resultant of two vectors P(arrowhead) and Q(arrowhead) has the magnitude of P(arrowhead). If...

The resultant of two vectors P(arrowhead) and Q(arrowhead) has the magnitude of P(arrowhead).

The resultant of two vectors P(arrowhead) and Q(arrowhead) has the magnitude of P(arrowhead). If P(arrowhead) is doubled, the new resultant will be inclined to Q at an angle =………………??

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### 1 Answer

The figure with the two cases is attached below.

The value of the resultants `R` is:

`R^2 = P^2+Q^2-2PQcos(a)`

`P^2 =P^2 +Q^2-2PQ*cos(a)`

`Q(Q-2P*cos(a)) =0`

`cos(a) = Q/(2P)`

`a =arccos(Q/(2P))`

The value of the resultant `R'` is

`R'^2 =4P^2 +Q^2 -4PQcos(a)`

`R'^2 =4P^2 +Q^2 -4PQ*(Q/2P) =4P^2 +Q^2-2Q^2 =4P^2-Q^2`

Now from the sinus theorem in the two triangles we have

`P/sin(a) =P/sin(b) ` , (thus `a =b` )

`(R')/sin(a) =(2P)/sin(b')`

by dividing the two relation above we obtain

`R/(R') =(1/2)*sin(b')/sin(b)` or

`sin(b')/sin(b) = (2R)/(R')`

`sin(b)/sin(b') = (R')/(2P)`

`sin(b)/sin(b') = sqrt(1-(Q/(2P))^2)`

`sin(b') =sin(b)/[sqrt(1-(Q/(2P))^2)]`

`b' = arcsin(sin(b)/[sqrt(1-(Q/(2P))^2]))`