Vector Algebra
Cross Product and Right Handed System
Grade None
Question:
<p>Since <strong>ā, c̄, b̄</strong> form a right handed system, find <strong>c̄</strong> given that <strong>b̄ = (0, 1, 0)</strong> and <strong>ā = (x, y, z)</strong>.</p><p>If <strong>ā</strong>, <strong>c̄</strong>, <strong>b̄</strong> form a right handed system and \(\vec{c} = \vec{b} \times \vec{a}\), then \(\vec{c}\) equals:</p>
<p>\(z\hat{i} - x\hat{k}\)</p>
<p>\(z\hat{i} + x\hat{k}\)</p>
<p>\(-z\hat{i} + x\hat{k}\)</p>
<p>\(x\hat{i} - z\hat{k}\)</p>
Step-by-Step Solution
Key Concept: In a right-handed system where ā, c̄, b̄ are in order, the relationship c̄ = b̄ × ā must hold (not b̄ × ā in arbitrary order). The cross product b̄ × ā with b̄ = (0,1,0) and ā = (x,y,z) gives the perpendicular vector that completes the right-handed triplet.
Step 1: For ā, c̄, b̄ to form a right-handed system in that order, we need c̄ = b̄ × ā. Step 2: Calculate b̄ × ā where b̄ = (0, 1, 0) and ā = (x, y, z): c̄ = (0, 1, 0) × (x, y, z) = | î ĵ k̂ | |0 1 0| |x y z| = î (1·z - 0·y) - ĵ (0·z - 0·x) + k̂ (0·y - 1·x) = î (z) - ĵ (0) + k̂ (-x) = (z, 0, -x) Verification: Check that ā·c̄ = xz + 0 - xz = 0 ✓ and b̄·c̄ = 0 + 0 + 0 = 0 ✓, confirming perpendicularity. ∴ Answer: c̄ = (z, 0, -x)
Correct Answer: A