Question

The cross product $(2\hat{i}+3\hat{j}+4\hat{k})\times (\hat{i}-\hat{j}+\hat{k})$ is .

• A. $7\hat{i}+2\hat{j}-5\hat{k}$
• B. $5\hat{i}-2\hat{j}+7\hat{k}$
• C. $2\hat{i}-7\hat{j}+5\hat{k}$

Answer 1

The correct option is A $7\hat{i}+2\hat{j}-5\hat{k}$

The two given vectors are
$u=2\hat{i}+3\hat{j}+4\hat{k}$
$v=\hat{i}-\hat{j}+\hat{k}$
Their cross product will be given by the following determinant
$u\times v=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 4\\ 1& -1& 1\end{array}\right|$
The second row of the determinant contains the coefficients of $\hat{i},\hat{j},\hat{k}$ in vector u and the third row contains the corresponding coefficients in vector v.
The determinant on evaluation will give:-
$\hat{i}\left(\right(3\times 1)-(-1\times 4\left)\right).\hat{j}\left(\right(2\times 1)-(1\times 4\left)\right)+\hat{k}\left(\right(2\times 1)-(3\times 1\left)\right)$
$=7\hat{i}+2\hat{j}-5\hat{k}$
So the sum of the coefficient of $\hat{i},\hat{j}$ and $\hat{k}$ will be $(7+2-5)=4.$