Question

If \( \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \), \( \mathbf{b} = \hat{i} + 2\hat{j} - 3\hat{k} \), and \( \mathbf{c} = 2\hat{i} + 3\hat{j} + 5\hat{k} \) are coplanar, then \( \lambda \) is the root of the equation

• A. \( x^2 + 2x = 6 \)
• B. \( x^2 + 2x = 4 \)
• C. \( x^2 + 3x = 6 \)
• D. \( x^2 + 3x = 4 \)

Hint:

For coplanarity, use the scalar triple product. If the result is zero, the vectors are coplanar.

Answer 1

The Correct Option is B

Step 1: Use the condition for coplanarity.
The vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar if the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \).
Step 2: Calculate the scalar triple product.
Compute \( \mathbf{b} \times \mathbf{c} \) and then take the dot product with \( \mathbf{a} \).
Step 3: Solve for \( \lambda \).
Solving the equation gives \( \lambda \) as the root of \( x^2 + 2x = 4 \).
Step 4: Conclusion.
The root of the equation is \( x^2 + 2x = 4 \).