• # question_answer Let $\mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}$ and $\mathbf{c}=\mathbf{i}+\mathbf{j}-2\mathbf{k}$ be three vectors. A vector in the plane of b and c whose projection on a is of magnitude $\sqrt{2/3}$ is [IIT 1993; Pb. CET 2004] A) $2\mathbf{i}+3\mathbf{j}-3\mathbf{k}$ B) $2\mathbf{i}+3\mathbf{j}+3\mathbf{k}$ C) $-\,2\mathbf{i}-\mathbf{j}+5\mathbf{k}$ D) $2\mathbf{i}+\mathbf{j}+5\mathbf{k}$

Solution :

• Any vector $\mathbf{r}$ in the plane of $\mathbf{b}$ and $\mathbf{c}$ is $\mathbf{r}=\mathbf{b}+t\mathbf{c}$ or $\mathbf{r}=(1+t)\mathbf{i}+(2+t)\mathbf{j}-(1+2t)\mathbf{k}$                    ......(i)
• Projection of $\mathbf{r}$ on $\mathbf{a}$ is $\sqrt{\left( \frac{2}{3} \right)}\Rightarrow \frac{\mathbf{r}\,.\,\mathbf{a}}{|\mathbf{a}|}=\sqrt{\left( \frac{2}{3} \right)}$
• or  $\frac{2(1+t)-(2+t)-(1+2t)}{\sqrt{6}}=\pm \sqrt{\left( \frac{2}{3} \right)}$
• \ $\,-t-1=\pm 2\Rightarrow t=-3,\,\,1$
• Projection in (i),we get
• $\therefore \,\mathbf{r}=-2\mathbf{i}-\mathbf{j}+5\mathbf{k}$ or $\mathbf{r}=2\mathbf{i}+3\mathbf{j}-3\mathbf{k}$.

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