Vector Algebra
Scalar Triple Product
Grade 12
Question:
<p>If [<strong>a</strong> × <strong>b</strong> <strong>b</strong> × <strong>c</strong> <strong>c</strong> × <strong>a</strong>] = λ[<strong>a</strong> <strong>b</strong> <strong>c</strong>]², then λ is equal to</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: The scalar triple product [u v w] = u·(v × w) can be manipulated using properties of cyclic permutations and vector identities. We need to recognize that [a × b b × c c × a] is a scalar triple product and relate it to [a b c]² using the Binet-Cauchy identity.
Step 1: Recall that the scalar triple product [u v w] = u·(v × w) = (u × v)·w. Step 2: Write [a × b b × c c × a] = (a × b)·((b × c) × (c × a)). Step 3: Use the vector triple product formula: (b × c) × (c × a) = (b × c · a)c - (b × c · c)a = ([b c a])c - 0 = [b c a]c = [a b c]c. Step 4: Substitute back: (a × b)·([a b c]c) = [a b c]((a × b)·c) = [a b c][a b c] = [a b c]^2. Step 5: Therefore [a × b b × c c × a] = 1·[a b c]^2. ∴ Answer: B
Correct Answer: B