Question

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

• A. 462
• B. 77
• C. 308
• D. 154

Hint:

Use vector identities and properties of dot and cross products to solve for unknowns in vector equations.

Answer 1

The Correct Option is C

- Since \( \vec{a} \times \vec{c} = \vec{a} \times \vec{b} \), the vector \( \vec{c} \) lies in the plane defined by \( \vec{a} \) and \( \vec{b} \). 
- Using the condition \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168 \).
- we solve for \( |\vec{c}|^2 \), yielding a maximum value of 308.