A vector perpendicular to 2 i+j+k and coplanar with i+2 j+k and i… - Vidyadip

MCQ

A vector perpendicular to $2 \hat{i}+\hat{j}+\hat{k}$ and coplanar with $\hat{i}+2 \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+2 \hat{k}$ is

✓
$5(\hat{ j }-\hat{ k })$
B
$\hat{ i }+7 \hat{ j }-\hat{ k }$
C
$5(\hat{ j }+\hat{ k })$
D
$2 \hat{ i }-7 \hat{ j }-\hat{ k }$

✓

Answer

Correct option: A.

$5(\hat{ j }-\hat{ k })$

(A) Let the vector be $a \hat{i}+b \hat{j}+c \hat{k}$.
It is perpendicular to $2 \hat{i}+\hat{j}+\hat{k}$.
$\therefore \quad 2 a+b+c=0$ ...(i)
The vector is coplanar with $\hat{i}+2 \hat{j}+\hat{k}$ and $\hat{ i }+\hat{ j }+2 \hat{ k }$
$\therefore\left|\begin{array}{ccc}a & b & c \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right|=0$
$\therefore \quad 3 a-b-c=0$ ...(ii)
On solving (i) and (ii), we get
$a=0, b=5, c=-5$
∴ The required vector is $5(\hat{ j }-\hat{ k })$

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