Vector Algebra
Cross Product of Vectors
Grade 12
Question:
<p><strong>Example 30.</strong> Let <strong>A</strong>, <strong>B</strong> and <strong>C</strong> be unit vectors. Suppose <strong>A</strong> · <strong>B</strong> = <strong>A</strong> · <strong>C</strong> = 0 and the angle between <strong>B</strong> and <strong>C</strong> is <strong>π/4</strong>. Then:</p>
<p>(a) <strong>A</strong> = ± √2 (<strong>B</strong> × <strong>C</strong>)</p>
<p>(b) <strong>A</strong> = ± √(1/2) (<strong>B</strong> × <strong>C</strong>)</p>
<p>(c) <strong>A</strong> = ± √3 (<strong>B</strong> + <strong>C</strong>)</p>
<p>(d) <strong>A</strong> = ± √3 (<strong>B</strong> × <strong>C</strong>)</p>
Step-by-Step Solution
Key Concept: A perpendicular to both B and C must be parallel to their cross product. The magnitude of the cross product determines the scaling factor.
Step 1: Since A · B = 0 ⇒ A ⊥ B and A · C = 0 ⇒ A ⊥ C Step 2: Therefore A is parallel to B × C . Since A is a unit vector perpendicular to both B and C : Step 3: A = ± B × C / | B × C | Step 4: Here | B × C | = | B || C | sin(π/4) = 1 · 1 · (1/√2) = 1/√2 Step 5: So A = ± ( B × C )/(1/√2) = ± √2 ( B × C ) ∴ Answer is (b).
Correct Answer: B