Vector Algebra
Coplanar and Non-coplanar Vectors
Grade 12
Question:
<p>If <strong>a</strong>, <strong>b</strong> and <strong>c</strong> are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(\mathbf{a} + 2\mathbf{b} + 3\mathbf{c}\), \(\lambda\mathbf{b} + 4\mathbf{c}\) and \((2\lambda - 1)\mathbf{c}\) are non-coplanar for</p>
<p>(a) no value of \(\lambda\)</p>
Step-by-Step Solution
Key Concept: Three vectors are coplanar if and only if their scalar triple product equals zero. Express the vectors in a coordinate system and compute the determinant.
Solution:
For the three vectors to be non-coplanar, their scalar triple product must be non-zero. The vectors are coplanar if and only if
\(\begin{vmatrix} 1 & 2 & 3 \\ 0 & \lambda & 4 \\ 0 & 0 & 2\lambda - 1 \end{vmatrix} = 0\)
Expanding: \(1 \cdot \lambda \cdot (2\lambda - 1) = \lambda(2\lambda - 1) = 0\)
This gives \(\lambda = 0\) or \(\lambda = \frac{1}{2}\)
For all other values of \(\lambda\), the vectors are non-coplanar. However, the question asks for which values they are always non-coplanar. Since there exist values of \(\lambda\) that make them coplanar, the answer is no value of \(\lambda\) ensures they are non-coplanar for all scenarios.
Correct Answer: A