Question

Let \( \mathbf{a} = 2\hat{i} - \hat{j} + 3\hat{k} \), \( \mathbf{b} = 3\hat{i} - 5\hat{j} + \hat{k} \), and \( \mathbf{c} \) be a vector such that \( \mathbf{a} \times \mathbf{c} = \mathbf{a} \times \mathbf{b} \) and \[ (\mathbf{a} + \mathbf{c}) \cdot (\mathbf{b} + \mathbf{c}) = 168. \] Then the maximum value of \( | \mathbf{c} |^2 \) is:

• A. 462
• B. 77
• C. 308
• D. 154

Hint:

When dealing with vector equations, use properties such as the cross product and dot product to set up relationships between the vectors and solve for unknowns.

Answer 1

The Correct Option is C

Step 1: The given vector equation \( \mathbf{a} \times \mathbf{c} = \mathbf{a} \times \mathbf{b} \) implies that \( \mathbf{c} \) lies in the plane formed by \( \mathbf{a} \) and \( \mathbf{b} \). We will use this condition to express \( \mathbf{c} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \). 
Step 2: The dot product equation \( (\mathbf{a} + \mathbf{c}) \cdot (\mathbf{b} + \mathbf{c}) = 168 \) provides a second condition to find \( \mathbf{c} \). Expand the dot product and solve for \( | \mathbf{c} |^2 \). 
Step 3: By solving these equations, we find that the maximum value of \( | \mathbf{c} |^2 \) is 308. Thus, the correct answer is (3).