Question

Let \(\vec{a} = 2\hat{i} - \hat{j} + \hat{k}\), \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\) and \(\vec{c} = \hat{i} + \hat{j} - 2\hat{k}\) be three vectors. A vector of the type \(\vec{b} + \lambda\vec{c}\) for some scalar \(\lambda\), whose projection on \(\vec{a}\) is of magnitude \(\sqrt{\frac{2}{3}}\) is :

• A. \(2\hat{i} + \hat{j} + 5\hat{k}\)
• B. \(2\hat{i} + 3\hat{j} - 3\hat{k}\)
• C. \(2\hat{i} - \hat{j} + 5\hat{k}\)
• D. \(2\hat{i} + 3\hat{j} + 3\hat{k}\)

Hint:

Write \(\vec{b}+\lambda\vec{c}\) in components, take its dot product with \(\vec{a}\) and divide by \(|\vec{a}|=\sqrt6\).

Answer 1

The Correct Option is B

\(\vec{b} + \lambda\vec{c} = (1+\lambda)\hat{i} + (2+\lambda)\hat{j} - (1+2\lambda)\hat{k}\).
\((\vec{b}+\lambda\vec{c})\cdot\vec{a} = 2(1+\lambda) - (2+\lambda) - (1+2\lambda) = -1-\lambda\).
\(|\vec{a}| = \sqrt{4+1+1} = \sqrt6\), so the magnitude of the projection is \(\frac{|1+\lambda|}{\sqrt6}\).
\(\frac{|1+\lambda|}{\sqrt6} = \sqrt{\frac23} \Rightarrow |1+\lambda| = \sqrt{\frac{2\cdot 6}{3}} = 2\), so \(\lambda = 1\) or \(\lambda = -3\).
For \(\lambda = 1\): \(\vec{b}+\vec{c} = 2\hat{i} + 3\hat{j} - 3\hat{k}\), which is option (2). (\(\lambda=-3\) gives \(-2\hat{i}-\hat{j}+5\hat{k}\), not listed.)
Answer: (2) \(2\hat{i} + 3\hat{j} - 3\hat{k}\)