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If $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$, $\mathbf{c}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\mathbf{d}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$, then $(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=$
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2880 Upvotes
Verified Answer
The correct answer is:
$8 \hat{\mathbf{i}}-36 \hat{\mathbf{j}}+60 \hat{\mathbf{k}}$
If $\mathbf{a} \times \mathbf{b}=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 2 & 1 & -3 \\ 1 & -2 & 3\end{array}\right|$
$=\hat{\mathbf{i}}(3-6)-\hat{\mathbf{j}}(6+3)+\hat{\mathbf{k}}(-4-1)=-3 \hat{\mathbf{i}}-9 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$
and $\mathbf{c} \times \mathbf{d}=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \mathbf{k} \\ -1 & 1 & -4 \\ 1 & 1 & 2\end{array}\right|$
$=\hat{\mathbf{i}}(2+4)-\hat{\mathbf{j}}(-2+4)+\hat{\mathbf{k}}(-1-1)=6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$
Now,
$\begin{aligned}
(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d}) & =\left|\begin{array}{ccc}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
-3 & -9 & -5 \\
6 & -2 & -2
\end{array}\right| \\
& =\hat{\mathbf{i}}(18-10)-\hat{\mathbf{j}}(6+30)+\hat{\mathbf{k}}(6+54) \\
& =8 \hat{\mathbf{i}}-36 \hat{\mathbf{j}}+60 \hat{\mathbf{k}}
\end{aligned}$
$=\hat{\mathbf{i}}(3-6)-\hat{\mathbf{j}}(6+3)+\hat{\mathbf{k}}(-4-1)=-3 \hat{\mathbf{i}}-9 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$
and $\mathbf{c} \times \mathbf{d}=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \mathbf{k} \\ -1 & 1 & -4 \\ 1 & 1 & 2\end{array}\right|$
$=\hat{\mathbf{i}}(2+4)-\hat{\mathbf{j}}(-2+4)+\hat{\mathbf{k}}(-1-1)=6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$
Now,
$\begin{aligned}
(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d}) & =\left|\begin{array}{ccc}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
-3 & -9 & -5 \\
6 & -2 & -2
\end{array}\right| \\
& =\hat{\mathbf{i}}(18-10)-\hat{\mathbf{j}}(6+30)+\hat{\mathbf{k}}(6+54) \\
& =8 \hat{\mathbf{i}}-36 \hat{\mathbf{j}}+60 \hat{\mathbf{k}}
\end{aligned}$
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