Question

If \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - \hat{j} + 4\hat{k} \), and \( \vec{c} = \hat{i} + \hat{j} + \hat{k} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then the value of \( \lambda \) is:

• A. \( \pm 1 \)
• B. 3
• C. 0
• D. \( -1 \)

Hint:

For perpendicular vectors, the dot product must be zero. Use this condition to solve for unknowns like \( \lambda \).

Answer 1

The Correct Option is B

For \( \vec{a} + \lambda \vec{b} \) to be perpendicular to \( \vec{c} \), the dot product of \( (\vec{a} + \lambda \vec{b}) \) and \( \vec{c} \) must be 0. The dot product condition is: \[ (\vec{a} + \lambda \vec{b}) \cdot \vec{c} = 0 \] First, express the vectors: \[ \vec{a} = \hat{i} + 2\hat{j} + \hat{k}, \quad \vec{b} = \hat{i} - \hat{j} + 4\hat{k}, \quad \vec{c} = \hat{i} + \hat{j} + \hat{k} \] Now, compute the dot product: \[ (\vec{a} + \lambda \vec{b}) \cdot \vec{c} = (\hat{i} + 2\hat{j} + \hat{k} + \lambda (\hat{i} - \hat{j} + 4\hat{k})) \cdot (\hat{i} + \hat{j} + \hat{k}) \] \[ = (\hat{i} + 2\hat{j} + \hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) + \lambda (\hat{i} - \hat{j} + 4\hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) \] \[ = (1 + 2 + 1) + \lambda (1 - 1 + 4) \] \[ = 4 + \lambda (4) \] \[ = 4 + 4\lambda \] For this to be 0, we have: \[ 4 + 4\lambda = 0 \] \[ \lambda = -1 \] Thus, \( \lambda = -1 \).