Question

The two vectors $(x^2-1)\hat{i}+(x+2)\hat{j}+x^2\hat{k}$ & $2\hat{i}-x\hat{j}+3\hat{k}$ are orthogonal.

• A. For no real value of x
• B. For $x=-1$
• C. For $x=\frac{1}{2}$
• D. For $x=-\frac{1}{2}$ & $x=1$

Answer 1

Answer: (D)

For $x=-\frac{1}{2}$ & $x=1$

$\left(x^2-1\right)\hat{i}+\left(x+2\right)\hat{j}+x^2\hat{k}and2\hat{i}-x\hat{j}+3\hat{k}\Rightarrow 2\left(x^2-1\right)-x\left(x+2\right)+3x^2=0\Rightarrow 2x^2-2-x^2-2x-3x^2=0\Rightarrow 5x^2-x^2-2x-2=0\Rightarrow 4x^2-2x-2=0\Rightarrow 2x^2-x-1=0\Rightarrow 2x^2-2x+x-1\Rightarrow 2x\left(x-1\right)+\left(x-1\right)=0\Rightarrow \left(2x+1\right)\left(x-1\right)=0\therefore x=\frac{-1}{2}\&x=1Ans.$