If a is a nonzero vector such that its projections on the vectors 2i-j+2k,i+2j-2k and k ar…

Question

If $\overset{	o }{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k},\hat{i}+2\hat{j}-2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\overset{	o }{a}$ is :

Accepted Answer

Projection of $\vec{a}$ on $\vec{v}$

$\begin{gathered} =\frac{\vec{a}\cdot \vec{v}}{|\vec{v}|}- \ \Rightarrow \frac{\vec{a}\cdot (2\hat{i}-\hat{j}+2\hat{k})}{3}=\frac{\vec{a}\cdot \hat{k}}{1}=\frac{\vec{a}\cdot (\hat{i}+2\hat{j}-2\hat{k})}{3} \ \Rightarrow \vec{a}\cdot (2\hat{i}-\hat{j}-\hat{k})=0\vec{a}\cdot (\hat{i}+2\hat{j}-5\hat{k})=0 \ \Rightarrow \vec{a}\perp (2\hat{i}-\hat{j}-\hat{k})(\hat{i}+2\hat{j}-5\hat{k}) \ \Rightarrow \vec{a}\parallel (2\hat{i}-\hat{j}-\hat{k})	imes (\hat{i}+2\hat{j}-5\hat{k}) \ \Rightarrow \vec{a}=\pm k|\begin{array}{ccc}\hat{i} \quad  -\hat{j} \quad  \hat{k} \ 2 \quad  -1 \quad  -1 \ 1 \quad  2 \quad  -5\end{array}|=\pm k(7\hat{i}+9\hat{j}-5\hat{k}) \ \Rightarrow \frac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}+5\hat{k}) \end{gathered}$

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