If $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}} = \lambda\widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$ and the orthogonal projection of $\overrightarrow{\mathbf{b}}$ on $\overrightarrow{\mathbf{a}}$ is $\frac{4}{3}\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right)then\ \lambda\ is\ equal\ to$

Question

Dr. Abubacker · Accepted Answer

(b)
$Given,\ \frac{\left( \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{a}} \right) \bullet \overrightarrow{\mathbf{a}}}{\left| \overrightarrow{\mathbf{a}} \right|^{2}} = \frac{4}{3}\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right)$
$\Longrightarrow \ \frac{\left\{ \left( \lambda\widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right).\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) \right\}\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right)}{(1 + 1 + 1)}$
$= \frac{4}{3}\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right)$
$\Longrightarrow (\lambda + 3 - 1)\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right) = 4\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right)$
$\Longrightarrow (\lambda + 2)\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right) = 4\left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right)$
On equating the coefficient of $\widehat{\dot{\mathbf{i}}}$, we get
$\lambda + 2 = 4 \Longrightarrow \lambda = 2$

Explanation

Reviewed by Dr. Abubacker

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