Question

If the vectors $c,a=x\hat{i}+y\hat{j}+z\hat{k}$ and $b=\hat{j}$ are such that $a,c,$ and $b$ form a right handed system. Then $c$ is?

• A. $z\hat{i}-x\hat{k}$
• B. $0$
• C. $y\hat{j}$
• D. $-z\hat{i}+x\hat{k}$

Answer 1

The correct option is A

$z\hat{i}-x\hat{k}$

Explanation for the correct option :

Find the vector $c$.

As the vectors $a,c,b$form a right handed system. So $c=b\times a$.

Now as $a=x\hat{i}+y\hat{j}+z\hat{k}$ and $b=\hat{j}$ so the cross product is given as:

$\begin{array}{rcl}b\times a& =& \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 0& 1& 0\\ x& y& z\end{array}\right|\\ & =& \hat{i}(z-0)-\hat{j}(0-0)+\hat{k}(0-x)\\ & =& z\hat{i}-x\hat{k}\end{array}$

So, the value of vector $c$ is $z\hat{i}-x\hat{k}$.

Hence, the correct option is A.