Vector Algebra
Scalar Triple Product
Grade 12
Question:
<p>The value of <span>\(|(p + q)\cos\theta + r|\)</span> is</p>
<p>(A) <span>\((1 + \cos\theta)\sqrt{1 - 2\cos\theta}\)</span></p>
<p>(B) <span>\(2\sin^2\frac{\theta}{2}|1 + 2\cos\theta|\)</span></p>
<p>(C) <span>\((1 - \sin\theta)\sqrt{1 + 2\cos\theta}\)</span></p>
<p>(D) None of the above</p>
Step-by-Step Solution
Key Concept: Calculate the determinant for the scalar triple product and simplify using half-angle formulas.
Step 1: From the equations derived, we have \([\mathbf{a}\mathbf{b}\mathbf{c}] = (1 - \cos^2\theta)(1 + 2\cos\theta)\) Step 2: Taking absolute value: \(v = |[\mathbf{a}\mathbf{b}\mathbf{c}]| = |1 - \cos^2\theta||1 + 2\cos\theta| = \sin^2\theta|1 + 2\cos\theta|\) Step 3: Using \(\sin^2\theta = 4\sin^2\frac{\theta}{2}\cos^2\frac{\theta}{2}\) or \(2\sin^2\frac{\theta}{2} = 1 - \cos\theta\) : \(v = 2\sin^2\frac{\theta}{2}|1 + 2\cos\theta|\) ∴ Answer is B .
Correct Answer: B