Resultant of two vectors `vecA and vecB` is of magnitude P, If `vecB` is reversed, then resultant is of magnitude Q. What is the value of `P^(2) + Q^(2)` ?

Options

A. `2(A^(2)+B^(2))`
B. `2(A^(2)-B^(2))`
C. `A^(2)-B^(2)`
D. `A^(2)+B^(2)`

Answer

A

Explanation

If `theta` is the angle between `vec A and vec B` then their resultant
`P = sqrt (A^(2) + B^(2) + 2AB cos theta)`
`P^(2) =A^(2) +B^(2) + 2AB cos theta" …(1)"`
and when `vec B` is reversed, the angle between them is `(180^(@) - theta)`
`therefore Q = sqrt (A^(2) +B^(2) +2AB cos (180^(@) - theta))`
`= sqrt (A^(2) + B^(2) -2AB cos theta)`
`therefore Q^(2) =A^(2) +B^(2) - 2AB cos theta" ...(2)"`
`therefore` From (1) and (2)
`P^(2) + Q^(2) = 2 (A^(2) + B^(2))" ...(3)"`