<p>If the vectors \(\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}\) and such that \(\vec{a}\), \(\vec{c}\) and \(\vec{b}\) form a right handed system, then \(\vec{c}\) is</p>
Step-by-Step Solution
Key Concept: In a right-handed system, if vectors a, c, b form the system in that order, then c = (b × a)/(|b||a|sin θ) or more directly, c must be perpendicular to both a and b such that a × c points in direction of b. The right-hand rule determines the unique direction: curl fingers from a to c, thumb points toward b.
Step 1: Understand the right-handed system condition. If vectors a, c, b form a right-handed system in that order, then by definition: a × c must point in the direction of b. Step 2: Since a × c is parallel to b, we have a × c = λb for some scalar λ > 0. For a proper right-handed orthonormal system, we need a × c = |a × c| · (b/|b|). Step 3: Since a and c are perpendicular in a right-handed system and assuming unit vectors: a × c = b (up to normalization). Step 4: Solve for c using the vector triple product: (a × c) × a = b × a, which gives c|a|^2 - a(a·c) = b × a. Since a ⊥ c (in orthonormal system), a·c = 0, so c = (b × a)/|a|^2. Step 5: For the standard right-handed system with unit vectors: c = (b × a)/(|a × b|) or equivalently c = b × a (normalized appropriately). ∴ Answer: A
Correct Answer: A