Computing u × v

Trigonometry · Axiom Academy

EXAMPLE Computing Cross Products Learn to find the cross product of two vectors using the determinant method and verify perpendicularity. Excellent work! You've successfully computed a cross product and verified perpendicularity. Here's what we learned: Determinant Setup: The cross product u × v is computed using a 3×3 determinant with i , j , k in the first row, components of u in the second row, and components of v in the third row. Expansion Method: Expand along the first row, remembering to alternate signs: + i term, - j term, + k term. Each term involves a 2×2 determinant. Geometric Property: The cross product u × v is perpendicular to both u and v . This is verified by showing that the dot products are zero. Right-Hand Rule: The direction of u × v follows the right-hand rule: point your fingers along u , curl them toward v , and your thumb points in the direction of u × v. Order Matters: Note that v × u = -(u × v), so the order of vectors in the cross product affects the direction of the result. The cross product is essential in physics and engineering for finding normal vectors, calculating torque, and determining areas of parallelograms. Practice this method to build confidence!

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