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If the scalar projection of the vectors $x \mathbf{i}-\mathbf{j}+\mathbf{k}$ on the vector $2 \mathbf{i}-\mathbf{j}+5 \mathbf{k}$ is $\frac{1}{\sqrt{30}}$ then value of $x$ is equal to
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The correct answer is:
$\frac{-5}{2}$
Projection of $x \mathbf{i}-\mathbf{j}+\mathbf{k}$ on $2 \mathbf{i}-\mathbf{j}+5 \mathbf{k}$
$=\frac{(x \mathbf{i}-\mathbf{j}+\mathbf{k})(2 \mathbf{i}-\mathbf{j}+5 \mathbf{k})}{\sqrt{4+1+25}}=\frac{2 x+1+5}{\sqrt{30}}$
But, given $\frac{2 x+6}{\sqrt{30}}=\frac{1}{\sqrt{30}} \Rightarrow 2 x+6=1 \Rightarrow x=\frac{-5}{2}$.
$=\frac{(x \mathbf{i}-\mathbf{j}+\mathbf{k})(2 \mathbf{i}-\mathbf{j}+5 \mathbf{k})}{\sqrt{4+1+25}}=\frac{2 x+1+5}{\sqrt{30}}$
But, given $\frac{2 x+6}{\sqrt{30}}=\frac{1}{\sqrt{30}} \Rightarrow 2 x+6=1 \Rightarrow x=\frac{-5}{2}$.
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