Vector Algebra
Vector triple product
Grade 12
Question:
<p>If \((\vec{a}\times\vec{b})\times\vec{c} = \vec{a}\times(\vec{b}\times\vec{c})\), where \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are any three vectors such that \(\vec{a}\cdot\vec{b}\neq 0\), \(\vec{b}\cdot\vec{c}\neq 0\), then \(\vec{a}\) and \(\vec{c}\) are</p>
<p>inclined at an angle of \(\pi/3\) between them.</p>
<p>inclined at an angle of \(\pi/6\) between them.</p>
<p>perpendicular.</p>
<p>parallel.</p>
Step-by-Step Solution
Key Concept: Use the vector triple product formula: (A×B)×C = (A·C)B - (B·C)A to expand both sides, then compare coefficients to find the relationship between vectors a and c.
Step 1: Apply vector triple product formula (A×B)×C = (A·C)B - (B·C)A Left side: (a×b)×c = (a·c)b - (b·c)a Right side: a×(b×c) = (a·c)b - (a·b)c Step 2: Set them equal: (a·c)b - (b·c)a = (a·c)b - (a·b)c Step 3: Simplify by canceling (a·c)b from both sides: -(b·c)a = -(a·b)c (b·c)a = (a·b)c Step 4: Rearrange to get: a/|a·b| = c/|b·c| (up to scalar multiples) This shows: a = λc where λ is a scalar constant ∴ Answer: a and c are parallel (or collinear)
Correct Answer: D