Vector Algebra
Scalar Triple Product
Grade 12
Question:
<p>Which of the following statements is/are correct? (a) If \(\mathbf{n} \cdot \mathbf{a} = 0, \mathbf{n} \cdot \mathbf{b} = 0\) and \(\mathbf{n} \cdot \mathbf{c} = 0\) for some non-zero vector \(\mathbf{n}\), then \([\mathbf{a} \mathbf{b} \mathbf{c}] = 0\)</p>
<p>(a) True - the vectors are coplanar</p>
<p>(b) False - the vectors need not be coplanar</p>
<p>(c) True only if \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are non-zero</p>
<p>(d) Cannot be determined</p>
Step-by-Step Solution
Key Concept: Vectors perpendicular to a common non-zero vector are coplanar.
If \(\mathbf{n} \cdot \mathbf{a} = 0, \mathbf{n} \cdot \mathbf{b} = 0, \mathbf{n} \cdot \mathbf{c} = 0\) for a non-zero vector \(\mathbf{n}\), then \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) all lie in the plane perpendicular to \(\mathbf{n}\). Therefore they are coplanar and \([\mathbf{a} \mathbf{b} \mathbf{c}] = 0\).
Correct Answer: A