Vector Algebra
Scalar Triple Product Properties
Grade None
Question:
<p>If <strong>u</strong>, <strong>v</strong> and <strong>w</strong> are three non-coplanar vectors, then \((\mathbf{u} + \mathbf{v} - \mathbf{w}) \cdot [(\mathbf{u} - \mathbf{v}) \times (\mathbf{v} - \mathbf{w})]\) is equal to</p>
<p>(a) 0</p>
<p>(b) \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\)</p>
<p>(c) \(\mathbf{u} \cdot (\mathbf{w} \times \mathbf{v})\)</p>
<p>(d) \(3\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\)</p>
Step-by-Step Solution
Key Concept: Expand the cross product using distributive property and apply scalar triple product rules: \([\mathbf{u} \mathbf{u} \mathbf{v}] = 0\) and cyclic permutations.
Solution:
\((\mathbf{u} + \mathbf{v} - \mathbf{w}) \cdot [(\mathbf{u} - \mathbf{v}) \times (\mathbf{v} - \mathbf{w})]\)
\(\Rightarrow (\mathbf{u} + \mathbf{v} - \mathbf{w}) \cdot [(\mathbf{u} \times \mathbf{v}) - (\mathbf{u} \times \mathbf{w}) + (\mathbf{v} \times \mathbf{w})]\)
\(= [\mathbf{u} \mathbf{u} \mathbf{v}] + [\mathbf{v} \mathbf{u} \mathbf{v}] - [\mathbf{w} \mathbf{u} \mathbf{v}] - [\mathbf{u} \mathbf{u} \mathbf{w}] - [\mathbf{v} \mathbf{u} \mathbf{w}] + [\mathbf{w} \mathbf{u} \mathbf{w}] + [\mathbf{u} \mathbf{v} \mathbf{w}] + [\mathbf{v} \mathbf{v} \mathbf{w}] - [\mathbf{w} \mathbf{v} \mathbf{w}]\)
\(= 0 + 0 - [\mathbf{u} \mathbf{v} \mathbf{w}] - 0 + [\mathbf{u} \mathbf{v} \mathbf{w}] + 0 + [\mathbf{u} \mathbf{v} \mathbf{w}] + 0 - 0\)
\(= [\mathbf{u} \mathbf{v} \mathbf{w}] = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\)
∴ Answer is (b)
Correct Answer: B