Question

If \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors and \( \lambda \) is a real number, then the vectors \( \vec{a} + 2\vec{b} + 3\vec{c} \), \( \lambda \vec{b} + 4\vec{c} \), and \( (2\lambda - 1)\vec{c} \) are non-coplanar for:

• A. no value of \( \lambda \).
• B. all except one value of \( \lambda \).
• C. all except two values of \( \lambda \).
• D. all values of \( \lambda \).

Hint:

To check if vectors are non-coplanar, form a matrix with their coefficients in a basis and compute the determinant. Non-zero determinant indicates linear independence.