First slide

Scalar triple product of vectors

Question

If vectors $\vec{A} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}, \vec{B} = \hat{i} + \hat{j} + 5 \hat{k} and \vec{C}$ form a left-handed system, then $\vec{C}$ is

Moderate

A

$11 \hat{i} - 6 \hat{j} - \hat{k}$
B

$- 11 \hat{i} + 6 \hat{j} + \hat{k}$
C

$11 \hat{i} - 6 \hat{j} + \hat{k}$
D

$- 11 \hat{i} + 6 \hat{j} - \hat{k}$

Solution

For $\vec{A}, \vec{B} and \vec{C}$ to form a left-handed system $[\vec{A} \vec{B} \vec{C}] < 0$
$\vec{A} \times \vec{B} = |\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 1 & 1 & 5 \end{array}| = 11 \hat{i} - 6 \hat{j} - \hat{k} - - - - - (i)$
(i) is satisfied by options (2) and (4).

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