Question

If $\overset{\rightarrow}{A} = i - 2j - 3k,\overset{\rightarrow}{B} = 2i + j - k,\overset{\rightarrow}{C} = i + 3j - 2k$, then $(\overset{\rightarrow}{A} \times \overset{\rightarrow}{B}) \times \overset{\rightarrow}{C}$ is

• A. $5( - i + 3j + 4k)$
• B. $4( - i + 3j + 4k)$
• C. $5( - i - 3j - 4k)$
• D. $4(i + 3j + 4k)$

Answer 1

Answer: (A)

$5( - i + 3j + 4k)$

Answer 2

\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & - 2 & - 3 \\ 2 & 1 & - 1 \end{matrix} \right| = \mathbf{i}(2 + 3) - \mathbf{j}( - 1 + 6) + \mathbf{k}(1 + 4)$$ $$= 5\mathbf{i} - 5\mathbf{j} + 5\mathbf{k}$$ Now $(\overrightarrow{A} \times \overrightarrow{B}) \times \overrightarrow{C} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & - 5 & 5 \\ 1 & 3 & - 2 \end{matrix} \right|$ $$= \mathbf{i}(10 - 15) - \mathbf{j}( - 10 - 5) + \mathbf{k}(15 + 5)$$ $= - 5\mathbf{i} + 15\mathbf{j} + 20\mathbf{k} = 5( - \mathbf{i} + 3\mathbf{j} + 4\mathbf{k})$.