Vector Algebra
Vector identities
Grade 12
Question:
<p>We have <span>\(\vec{b} \times \vec{c} = \vec{b} \times \vec{d}\)</span>. Then <span>\(\vec{d}\)</span> equals:</p>
<p>\(\vec{c} + \left(\dfrac{\vec{a}\cdot\vec{c}}{\vec{a}\cdot\vec{b}}\right)\vec{b}\)</p>
<p>\(\vec{c} - \left(\dfrac{\vec{a}\cdot\vec{c}}{\vec{a}\cdot\vec{b}}\right)\vec{b}\)</p>
<p>\(\vec{b} + \left(\dfrac{\vec{a}\cdot\vec{b}}{\vec{a}\cdot\vec{c}}\right)\vec{c}\)</p>
<p>\(\vec{b} - \left(\dfrac{\vec{a}\cdot\vec{b}}{\vec{a}\cdot\vec{c}}\right)\vec{c}\)</p>
Step-by-Step Solution
Key Concept: The equation b × c = b × d means b × (c - d) = 0, which occurs when (c - d) is parallel to b, not when c = d. This allows d to differ from c by any vector along b's direction.
Step 1: Start with the given equation: b × c = b × d Step 2: Rearrange using distributive property: b × c - b × d = 0 Step 3: Factor out b : b × ( c - d ) = 0 Step 4: The cross product equals zero when vectors are parallel. Therefore: c - d = λ b for some scalar λ Step 5: This means: d = c - λ b or equivalently d = c + μ b (where μ = -λ) Step 6: The general form shows d differs from c by a component parallel to b . ∴ Answer: B (Typically expressed as d = c + λ b or d = c + t b for scalar t)
Correct Answer: B