Discussion Forum

Let a and b be two unit vectors such that a.b=0. For some x,y $\in$ R, let c=xa+yb+ (a× b) . If  $\mid c\mid=2$ and the vector c is inclined at the same angle $\alpha$ to both a and b, then the value of $8\cos^{2}\alpha$  is

Answer:

Explanation:

We have

  $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+\overrightarrow{a}\times\overrightarrow{b}$   and  $\overrightarrow{a}.\overrightarrow{b}=0$

    $\mid\overrightarrow{a}\mid=\mid\overrightarrow{b}\mid=1$ and $\mid\overrightarrow{c}\mid=2$

   Also , given $\overrightarrow{c}$ is inclined on $\overrightarrow{a}$ and $\overrightarrow{b}$ with same angle $\alpha$

$\therefore$     $\overrightarrow{a}$ . $\overrightarrow{c}$ = $x\mid\overrightarrow{a}\mid^{2}+y(\overleftarrow{a}.\overrightarrow{b})+\overrightarrow{a}.(\overrightarrow{a}\times\overrightarrow{b})$

  $\mid\overrightarrow{a}\mid^{}.\mid\overrightarrow{c}\mid^{}\cos\alpha=x+0+0$ 

                            x= $2\cos\alpha$

Similarly,

       $\mid\overrightarrow{b}\mid^{}.\mid\overrightarrow{c}\mid^{}\cos\alpha=0+y+0$

                     y = $2\cos\alpha$

 $\mid\overrightarrow{c}\mid^{2}=x^{2}+y^{2}+\mid\overrightarrow{a}\times\overrightarrow{b}\mid^{2}$

   $4= 8\cos^{2}\alpha+\mid a\mid^{2}\mid b\mid^{2}\sin^{2}90^{0}$

    $4= 8\cos ^{2}\alpha+1$

$\Rightarrow$ $8\cos^{2}\alpha=3$