Question

A vector perpendicular to the plane containing the points $A(1,-1,2)$, $B(2,0,-1)$, $C(0,2,1)$ is

• A. $4\hat{i}+8\hat{j}-4\hat{k}$
• B. $8\hat{i}+4\hat{j}+4\hat{k}$
• C. $3\hat{i}+\hat{j}+2\hat{k}$
• D. $\hat{i}+\hat{j}-\hat{k}$

Answer 1

The correct option is B $8\hat{i}+4\hat{j}+4\hat{k}$

we know that a vector perpendicular to the plane containing the point $A,B,C$ is given by
$A\times B+B\times C+C\times A$
Given, $A=\hat{i}-\hat{j}+2\hat{k},B=2\hat{i}-0\hat{j}-\hat{k},C=0\hat{i}+2\hat{j}+\hat{k}$
$A\times B=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -1& 2\\ 2& 0& -1\end{array}\right|=\hat{i}+5\hat{j}+2\hat{k}$
$B\times C=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 0& -1\\ 0& 2& 1\end{array}\right|=2\hat{i}-2\hat{j}+4\hat{k}$
$C\times A=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 0& 2& 1\\ 1& -1& 2\end{array}\right|=5\hat{i}+\hat{j}-2\hat{k}$
$A\times B+B\times C+C\times A=8\hat{i}+4\hat{j}+4\hat{k}$