Mathematics

Problem

Two non-collinear vectors

\mathbf{P}

and

\mathbf{Q}

lie in the x-y plane. The magnitude of

\mathbf{P}

P

and that of

\mathbf{Q}

Q

. The angle from

\mathbf{P}

\mathbf{Q}

, measured through the smaller rotation in the plane, is

\theta

. a. (i) Resolve

\mathbf{P}

and

\mathbf{Q}

into their respective x and y components in terms of

P

Q

and

\theta

. b. (ii) By adding the components obtained in part (i), write expressions for the components

R_x

and

R_y

of the resultant vector

\mathbf{R}=\mathbf{P}+\mathbf{Q}

. c. (iii) Eliminating

R_x

and

R_y

, derive the general expression

R=\sqrt{P^{2}+Q^{2}+2PQ\cos\theta}

for the magnitude

R

of the resultant. d. (iv) Obtain an expression for the angle

\alpha

that

\mathbf{R}

makes with

\mathbf{P}

in terms of

P

Q

and

\theta

. Clearly state each step used in the derivation.