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 =………………??

valentin68 | College Teacher | (Level 3) Associate Educator

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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]))`

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