Vector Algebra
Scalar and Vector Products
Grade 12
Question:
<p>Which of the following expressions are meaningful?</p><p>(a) \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\)</p><p>(b) \((\mathbf{u} \cdot \mathbf{v}) \cdot \mathbf{w}\)</p><p>(c) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\)</p><p>(d) \(\mathbf{u} \times (\mathbf{v} \cdot \mathbf{w})\)</p>
<p>(a) \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\)</p>
<p>(b) \((\mathbf{u} \cdot \mathbf{v}) \cdot \mathbf{w}\)</p>
<p>(c) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\)</p>
<p>(d) \(\mathbf{u} \times (\mathbf{v} \cdot \mathbf{w})\)</p>
Step-by-Step Solution
Key Concept: Understand which operations produce vectors vs scalars: dot product yields scalar, cross product requires two vectors, scalar multiplication of vector yields vector.
Analysis: (i) Since \(\mathbf{v} \times \mathbf{w}\) is a vector, therefore \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) is a scalar quantity. ∴ (a) is meaningful. (ii) \((\mathbf{u} \cdot \mathbf{v})\) is a scalar. ∴ \((\mathbf{u} \cdot \mathbf{v}) \cdot \mathbf{w}\) is not meaningful (cannot take dot product of scalar with vector). (iii) \((\mathbf{u} \cdot \mathbf{v})\) is a scalar. So \((\mathbf{u} \cdot \mathbf{v})\mathbf{w}\) is a scalar multiple of \(\mathbf{w}\). ∴ (c) is meaningful. (iv) \((\mathbf{v} \cdot \mathbf{w})\) is a scalar. So \(\mathbf{u} \times (\mathbf{v} \cdot \mathbf{w})\) is not meaningful as cross product is defined for two vectors, not a vector and scalar. ∴ Correct answers are (a, c)
Correct Answer: A, C