Vector Algebra
Scalar and Vector Triple Products
Grade 12
Question:
<p>If <strong>a</strong>, <strong>b</strong>, <strong>c</strong>, and <strong>d</strong> are vectors such that <strong>(a × b) × (c × d) · (a × d) = 0</strong>, then which of the following may be true?</p>
<p>(a) <strong>a</strong>, <strong>b</strong>, <strong>c</strong> and <strong>d</strong> are necessarily coplanar</p>
<p>(b) <strong>a</strong> lies in the plane of <strong>c</strong> and <strong>d</strong></p>
<p>(c) <strong>b</strong> lies in the plane of <strong>a</strong> and <strong>d</strong></p>
<p>(d) <strong>c</strong> lies in the plane of <strong>a</strong> and <strong>d</strong></p>
Step-by-Step Solution
Key Concept: Use vector triple product expansion and scalar triple product properties to determine coplanarity conditions.
Step 1: Expand using the vector triple product formula: (a × b) × (c × d) = [a c d]b - [b c d]a Step 2: Take dot product with (a × d) : ([a c d][d a d] - [b c d][a a d]) = 0 Step 3: Since [d a d] = 0 , we get [b c d][a a d] = 0 Step 4: This implies either b or c must lie in the plane of a and d . ∴ Options (b), (c), (d) are correct.
Correct Answer: b, c, d