If a¯=i¯−j¯−k¯, b¯=λi¯−3j¯+k¯ and the orthogonal projection of b¯ on a¯ is 43(i¯−j¯−k¯) then λ= ?

Given a¯=i¯−j¯−k¯, b¯=λi¯−3j¯+k¯⇒(b¯⋅a¯)−a¯|a¯|2=34(l¯−j¯−k¯)⇒λ+3−13⋅(i¯−j¯−k¯)=34(i¯−j¯−k¯)⇒λ=2

If a¯=i¯−j¯−k¯, b¯=λi¯−3j¯+k¯ and the orthogonal projection of b¯ on a¯ is 43(i¯−j¯−k¯) then λ= ?

Given a¯=i¯−j¯−k¯, b¯=λi¯−3j¯+k¯⇒(b¯⋅a¯)−a¯|a¯|2=34(l¯−j¯−k¯)⇒λ+3−13⋅(i¯−j¯−k¯)=34(i¯−j¯−k¯)⇒λ=2

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If $\bar{a} = \bar{i} - \bar{j} - \bar{k}, \bar{b} = λ \bar{i} - 3 \bar{j} + \bar{k}$ and the orthogonal projection of $\bar{b}$ on $\bar{a}$ is $\frac{4}{3} (\bar{i} - \bar{j} - \bar{k})$ then $λ =$ ?

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answer is B.

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Detailed Solution

Given $\bar{a} = \bar{i} - \bar{j} - \bar{k}, \bar{b} = λ \bar{i} - 3 \bar{j} + \bar{k}$
$\begin{array}{l} \Rightarrow \frac{(\bar{b} \cdot \bar{a})^{- \bar{a}}}{| \bar{a} |^{2}} = \frac{3}{4} (\bar{l} - \bar{j} - \bar{k}) \\ \Rightarrow (\frac{λ + 3 - 1}{3}) \cdot (\bar{i} - \bar{j} - \bar{k}) = \frac{3}{4} (\bar{i} - \bar{j} - \bar{k}) \\ \Rightarrow λ = 2 \end{array}$