Question

Consider the vectors $\overline{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\overline{b}=4\hat{i}-4\hat{j}+7\hat{k}$.Find the vector perpendicular to both the vectors.

• A. $-10\hat{i}-3\hat{j}+4\hat{k}$
• B. $-10\hat{i}+3\hat{j}+4\hat{k}$
• C. $10\hat{i}-3\hat{j}+4\hat{k}$
• D. None of the above

Answer 1

The correct option is A $-10\hat{i}-3\hat{j}+4\hat{k}$

Given $\overline{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\overline{b}=4\hat{i}-4\hat{j}+7\hat{k}$

Let $\overrightarrow{a}$ be the vector perpendicular to $\overrightarrow{a}$ and $\overrightarrow{b}$

$\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 1\\ 4& -4& 7\end{array}\right|=\hat{i}(-2\times 7+4\times 1)-\hat{j}(1\times 7-1\times 4)+\hat{k}(-4\times 1+2\times 4)=\hat{i}(-10)+\hat{j}(-3)+\hat{k}\left(4\right)=-10\hat{i}-3\hat{j}+4\hat{k}$