Vector Algebra
Scalar Triple Product
Grade None
Question:
<p>Volume of parallelopiped with edges <strong>a</strong>, <strong>b</strong> and <strong>c</strong> is</p>
<p>(A) <span>\(p + (q + r)\cos\theta\)</span></p>
<p>(B) <span>\((p + q + r)\cos\theta\)</span></p>
<p>(C) <span>\(2p - (q + r)\cos\theta\)</span></p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Use scalar triple product properties and take dot products of the given vector equation with each of the three vectors.
Step 1: We have \(\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c}\) Step 2: Taking dot product with a : \(\mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) + \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = p|\mathbf{a}|^2 + q(\mathbf{a} \cdot \mathbf{b}) + r(\mathbf{a} \cdot \mathbf{c})\) Step 3: Since \(|\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = 1\) and angle between any two vectors is \(\theta\) : \([\mathbf{a}\mathbf{b}\mathbf{c}] = p + q\cos\theta + r\cos\theta = p + (q + r)\cos\theta\) ∴ Answer is A .
Correct Answer: A