10. vector algebra — ગણિત ધોરણ 12 સાયન્સ — Question

and $\vec{c}=x \hat{i}+(x-2) \hat{j}-\hat{k}$

If $\vec{c}$ lies in the plane of $\vec{a}$ and $\vec{b}$

then $[\vec{a} \vec{b} \vec{c}]=0$

ie.e. $\left|\begin{array}{ccc}{1} & {1} & {1} \\ {1} & {-1} & {2} \\ {x} & {(x-2)} & {-1}\end{array}\right|=0$

$\Rightarrow 1[1-2(x-2)]-1[-1-2 x]$$+1[x-2+x]=0$

$\Rightarrow 1-2 x+4+1+2 x+2 x-2=0$

$\Rightarrow 2 x=-4 \quad \Rightarrow x=-2$