If a = i + 2 j + k , b = 2 i + j and c = 3 i - 4 j - 5 k , then f… - Vidyadip

Question

$\text{If}\overrightarrow{\text{a}} = \hat{\text{i}} + 2\hat{\text{j}} + \hat{\text{k}}, \overrightarrow{\text{b}} = 2\hat{\text{i}} + \hat{\text{j}}$ and $\overrightarrow{\text{c}} = 3\hat{\text{i}} - 4\hat{\text{j}} - 5\hat{\text{k}},$ then find a unit vector perpendicular to both of the vectors $(\overrightarrow{\text{a}} - \overrightarrow{\text{b}}) \text{and} \overrightarrow{\text{(c}} - \overrightarrow{\text{b}}). $

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Answer

$\overrightarrow{\text{a}} - \overrightarrow{\text{b}} = -\hat{\text{-i}} + \hat{\text{j}} + \hat{\text{k}}; \overrightarrow{\text{c}} - \overrightarrow{\text{b}} = \hat{\text{i}} - 5\hat{\text{j}} - 5\hat{\text{k}}$
$(\overrightarrow{\text{a}} - \overrightarrow{\text{b}}) \times(\overrightarrow{\text{c}} - \overrightarrow{\text{b}}) = \begin{vmatrix} \hat{\text{i}} & \hat{\text{j}} & \hat{\text{k}} \\ -1 & 1 & 1 \\ 1 & -5 & -5 \end{vmatrix} = - 4\hat{\text{j}} + 4\hat{\text{k}} $
$\therefore$ Unit vector perpendicular to both of the vectors $ = \frac{\hat{\text{j}}}{\sqrt{2}} + \frac{\hat{\text{k}}}{\sqrt{2}}$

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