Computing u × v for u = <1, 2, 3> and v = <2, -1, 1>

Calculus 3 · Axiom Academy

Expand a determinant to find the cross product, then verify it is perpendicular to both vectors Given and , compute the cross product using cofactor expansion of the determinant. Then verify the result is perpendicular to both and , and find its magnitude. The cross product is the determinant with in the top row, the components of in the middle row, and the components of in the bottom row. Nice work! You computed a cross product from the determinant and confirmed its defining geometric properties. Here is what to carry forward: Determinant setup: comes from a determinant with in the first row, in the second, and in the third. Cofactor expansion: Expand along the first row, alternating the signs +, -, + across the components. Perpendicularity: is perpendicular to both vectors — both and equal 0 . Geometric meaning: that magnitude is the area of the parallelogram formed by and , and the direction follows the right-hand rule. The cross product is a workhorse of vector calculus — it produces normal vectors to surfaces and drives quantities like torque and angular momentum in physics.

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