Vector product
Objective. Use cross products for normals and area.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The vector (cross) product a×b of two vectors produces a vector perpendicular to both a and b, with magnitude |a||b|sin θ equal to the area of the parallelogram they span. In component form, a×b is computed using the determinant-style expansion across the i, j, k unit vectors, giving (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). This is used to find normal vectors to planes and to calculate areas of triangles and parallelograms directly from vertex coordinates. For example, i×j = k follows directly from the right-hand rule built into this formula.
Worked example
Find the vector product a × b where a = (1,2,3) and b = (0,1,4), and hence state its magnitude exactly.
- 1i-component: (2×4)−(3×1) = 8−3 = 5
- 2j-component: −[(1×4)−(3×0)] = −(4−0) = −4
- 3k-component: (1×1)−(2×0) = 1−0 = 1
- 4So a×b = (5,−4,1), with magnitude √(25+16+1) = √42
Answer: a×b = (5, −4, 1), magnitude √42
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Practice
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Start practisingCommon mistake. Students compute the cross product as a scalar by multiplying magnitudes and sin θ only, forgetting it is a vector with direction perpendicular to both inputs. Always give a cross product as a vector with three components, not a single number.
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