Question

Determine a vector product of \[\vec A\] = \[\hat i\] + \[\hat j\] + \[\hat k\] and \[\vec B\] = -3\[\hat i\] + \[\hat j\] - 2\[\hat k\].

Options

• 3\[\hat i\] - \[\hat j\] + 4\[\hat k\]

• -3\[\hat i\] + \[\hat j\] + 4\[\hat k\]

• 3\[\hat i\] + \[\hat j\] - 4\[\hat k\]

• -3\[\hat i\] - \[\hat j\] + 4\[\hat k\]

Answer 1

-3\[\hat i\] - \[\hat j\] + 4\[\hat k\]

Explanation:

The cross product is computed using the determinant of a 3 × 3 matrix formed by unit vectors and the components of \[\vec A\] and \[\vec B\], giving

\[\vec{A}\times\vec{B}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ -3 & 1 & -2 \end{vmatrix}\]

= \[\hat{i}(1\cdot(-2)-1\cdot1)-\hat{j}(1\cdot(-2)-1\cdot(-3))+\hat{k}(1\cdot1-1\cdot(-3))\]

= \[\hat{i}(-3)-\hat{j}(1)+\hat{k}(4)=-3\hat{i}-\hat{j}+4\hat{k}\].