Question

Given the two vectors $i-j$ and $i+2j,$ the unit vector coplanar with the two perpendicular to that first is$?$

• A. $\frac{(i+k)}{\sqrt{2}}$
• B. $\frac{(2i+j)}{\sqrt{5}}$
• C. $\pm \frac{(i+j)}{\sqrt{2}}$
• D. none of these

Answer 1

The correct option is C

$\pm \frac{(i+j)}{\sqrt{2}}$

Explanation for the correct option:

Given, The two vectors are $i-jandi+2j$

The unit vector is coplanar with the given vectors and lie in $XY$ plane.

$a=x_i+y_j$

Now, a is perpendicular to $\left(i-j\right)$,

so$\left(x_i+y_j\right).\left(i-j\right)=0$

Thus,$$\begin{gathered} x-y=0 \\ x=y \\ a=xi+xj \\ \left|a\right|=\sqrt{\left(x^2+y^2\right)} \\ =x\sqrt{2} \end{gathered}$$

$$Hence , the unit vector a

$$\begin{gathered} =\frac{a}{\left|a\right|} \\ =\frac{x\left(i+j\right)}{x\sqrt{2}} \\ \mathbf{=}\frac{\left(i+j\right)}{\sqrt{\mathbf{2}}} \end{gathered}$$

Hence option(C) is correct.