Let two non-collinear unit vectors and form an acute angle

MATHEMATICS · Single correct · CBSE · Mathematics · Vector Algebra · Vector Algebra

Problem

Let two non-collinear unit vectors

\hat{a}

and

\hat{b}

form an acute angle. A point

P

moves so that at any time

t

the position vector

\vec{OP}

(where

O

is the origin) is given by

\hat{a}\cos t + \hat{b}\sin t

. When

P

is farthest from the origin

O

, let

M

be the length of

\vec{OP}

and

\hat{u}

be the unit vector along

\vec{OP}

. (1)

\hat{u} = \frac{\hat{a} + \hat{b}}{|\hat{a} + \hat{b}|}

and

M = (1 + \hat{a} \cdot \hat{b})^{1/2}

(2)

\hat{u} = \frac{\hat{a} - \hat{b}}{|\hat{a} - \hat{b}|}

and

M = (1 + \hat{a} \cdot \hat{b})^{1/2}

(3)

\hat{u} = \frac{\hat{a} + \hat{b}}{|\hat{a} + \hat{b}|}

and

M = (1 + 2\hat{a} \cdot \hat{b})^{1/2}

(4)

\hat{u} = \frac{\hat{a} - \hat{b}}{|\hat{a} - \hat{b}|}

and

M = (1 + 2\hat{a} \cdot \hat{b})^{1/2}