Let a = i + 4 j +2 k , b = 3 i - 2 j +7 k and c = 2 i - j + 4 k F… - Vidyadip

Question

Let $\overrightarrow{\text{a}} = \hat{\text{i}} + 4\hat{\text{j}} +2\hat{\text{k}}, \overrightarrow{\text{b}} = 3\hat{\text{i}} - 2\hat{\text{j}} +7\hat{\text{k}}$ and $\overrightarrow{\text{c}} = 2\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ Find a vector $\overrightarrow{\text{d}}$ which is perpendicular to both $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}}\text{and} \overrightarrow{\text{c}} . \overrightarrow{\text{d}} = 27.$

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Answer

$\text{Writing} \overrightarrow{\text{d}} = \lambda\bigg(\overrightarrow{\text{a}}\times\overrightarrow{\text{b}}\bigg)$
$= \lambda \begin{vmatrix} \hat{\text{i}} & \hat{\text{j}} & \hat{\text{k}} \\ 1 & 4 & 2 \\ 3 & -2 & 7 \end{vmatrix} $
$ = \lambda \bigg(32 \hat{\text{i}} - \hat{\text{j}} - 14\hat{\text{k}}\bigg)\dots\dots\dots\dots\text{(1)}$
$\overrightarrow{\text{c}}. \overrightarrow{\text{d}} = 27$
$\bigg(2\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}\bigg).\lambda\bigg(32\hat{\text{i}} - \hat{\text{j}} + 14\hat{\text{k}}\bigg) = 27$
$9\lambda = 27$
$\lambda = 3$
$\therefore\overrightarrow{\text{d}} = \bigg(96\hat{\text{i}} - \hat{\text{3j}} + 42\hat{\text{k}}\bigg)$

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