Calculate the cross product (vector product) of two 3D vectors. Find the resultant vector, magnitude, angle between vectors, and parallelogram area with step-by-step solutions.
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Calculate the cross product of two 3D vectors instantly. Our calculator shows the resultant vector in component form and i,j,k notation, computes magnitude, angle between vectors, and the area of the parallelogram formed by the vectors. Includes step-by-step solutions using the determinant method.
The cross product (vector product) of two vectors a and b produces a third vector c that is perpendicular to both input vectors. Unlike the dot product which gives a scalar, the cross product gives a vector. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, calculated as |a||b|sin(θ) where θ is the angle between them. The direction of the result follows the right-hand rule.
Cross Product Formula
a × b = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)kCross product calculations involve multiple terms with negative signs. Our calculator ensures accuracy every time.
Essential for calculating torque (τ = r × F), magnetic force (F = qv × B), and angular momentum.
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The magnitude |a × b| gives the area of the parallelogram formed by vectors a and b.
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The dot product (a · b) produces a scalar (single number) and measures how much two vectors point in the same direction. The cross product (a × b) produces a vector that is perpendicular to both inputs. Dot product uses cosine of the angle; cross product uses sine.
Point your fingers in the direction of vector a, curl them toward vector b, and your thumb points in the direction of a × b. For i × j = k: point fingers along x-axis, curl toward y-axis, thumb points along z-axis.
Yes, treat 2D vectors as 3D with z = 0. The cross product of two 2D vectors gives a vector along the z-axis only, essentially a scalar representing the signed area. Use our 2D mode for this calculation.
No, cross product is anti-commutative: a × b = -(b × a). Swapping the order reverses the direction of the result. This is why order matters when calculating torque or magnetic force.
Set up a 3×3 determinant with i, j, k in the first row, components of a in the second row, and components of b in the third row. Expand along the first row to get: i(a₂b₃-a₃b₂) - j(a₁b₃-a₃b₁) + k(a₁b₂-a₂b₁).
A zero cross product means the vectors are parallel (or anti-parallel). Since the magnitude equals |a||b|sin(θ), when θ = 0° or 180°, sin(θ) = 0, making the cross product zero.