Discussion Forum

1)

Let u be a vector coplanar with the vectors $a=2\hat{i}+3\hat{j}+\hat{k}$ and  $b=\hat{j}+\hat{k}$

.If u is perpendicular to a and u. b=24, then $\mid u\mid ^{2}$ is equal to 

Answer:

Option A

Explanation:

Key Idea  If any vector x is coplanar with the vector y and z then x=λy +μz Here, u is coplanar with a and b.

$\therefore$     u= λa+µb

  Dot product with a, we get

 u.a= λ (a.a)+µ (b.a)

$\Rightarrow$    0= 14λ+2 µ  ........(i)

    $[\therefore a= 2\hat{i}+3\hat{j}-\hat{k},b=\hat{j}+\hat{k},u.a=0]$

  Dot product with b, we get 

        $u.b=\lambda (a.b)+\mu (b.b)$

     $24=2\lambda+2\mu$   .....(ii)   $[\because u.b=24]$

Solving Eqs. (i) and(ii), we get

    λ =-2, μ = 14

Dot product with u, we get

$\mid u\mid^{2}=\lambda (u.a)+\mu (u.b)$

$\mid u\mid^{2}=-2 (0)+14 (24)$

$\Rightarrow$       $\mid u\mid^{2}=-336$