Let a = i +2 j + k , b =3 i -3 j +3 k , c =2 i - j +2 k and d be … - Vidyadip

Question

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}}$ be a vector such that $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}=4$. Then $|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{d}})|^{2}$ is equal to __________ .

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Answer

128
$\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}$-and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}=4$
$\Rightarrow \overrightarrow{\mathrm{d}}=\lambda(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}})=\lambda(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$
$\because \vec{a} \cdot \vec{d}=4 \Rightarrow \lambda=-2$
Also. $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{d}}|^{2}+|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}|\overrightarrow{\mathrm{~d}}|^{2}$
$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{d}}|^{2}=6 \times 4 \times 6-16=128$

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