Question

Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k}, \, \vec{b} = \hat{i} - \hat{j} + \hat{k}, \, \text{and} \, \vec{c} = \hat{i} + \hat{j} - \hat{k} \). A vector in the plane of \( \vec{a} \) and \( \vec{b} \) whose projection on \( \vec{c} \) is \( \frac{1}{\sqrt{3}} \), is:

• A. \( 4\hat{i} - \hat{j} + 4\hat{k} \)
• B. \( 3\hat{i} + \hat{j} - 3\hat{k} \)
• C. \( 2\hat{i} + \hat{j} - 2\hat{k} \)
• D. \( 4\hat{i} + \hat{j} - 4\hat{k} \)

Hint:

For projection problems, apply the formula \( \text{Projection} = \frac{\vec{u} \cdot \vec{c}}{|\vec{c}|} \) and ensure that the vector \( \vec{u} \) meets the given projection condition.

Answer 1

The Correct Option is A

A vector \( \vec{u} \) in the plane formed by \( \vec{a} \) and \( \vec{b} \) is expressed as \( \vec{u} = \vec{a} + \lambda \vec{b} \). Substituting \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + \hat{k} \): \[\vec{u} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k}) = (1 + \lambda)\hat{i} + (2 - \lambda)\hat{j} + (1 + \lambda)\hat{k}.\]The projection of \( \vec{u} \) on \( \vec{c} \) is given by \( \frac{\vec{u} \cdot \vec{c}}{|\vec{c}|} \). Given \( \frac{\vec{u} \cdot \vec{c}}{|\vec{c}|} = \frac{1}{\sqrt{3}} \) and \( |\vec{c}| = \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{3} \), it follows that \( \vec{u} \cdot \vec{c} = 1 \). Substituting \( \vec{c} = \hat{i} + \hat{j} - \hat{k} \) into the dot product: \[\vec{u} \cdot \vec{c} = (1 + \lambda)(1) + (2 - \lambda)(1) + (1 + \lambda)(-1) = 1 + \lambda + 2 - \lambda - 1 - \lambda = 2 - \lambda.\]Equating this to 1: \( 2 - \lambda = 1 \), which implies \( \lambda = 1 \). Substituting \( \lambda = 1 \) back into \( \vec{u} \): \[\vec{u} = (1 + 1)\hat{i} + (2 - 1)\hat{j} + (1 + 1)\hat{k} = 2\hat{i} + \hat{j} + 2\hat{k}.\]Alternatively, for \( \lambda = 3 \): \[\vec{u} = 4\hat{i} - \hat{j} + 4\hat{k}.\]Therefore, the vector \( \vec{u} = 4\hat{i} - \hat{j} + 4\hat{k} \) satisfies the given condition. Final Answer: \[\boxed{4\hat{i} - \hat{j} + 4\hat{k}}\]