What is the cross product of two vectors?

The cross product (also called vector product) of two 3D vectors a and b is a vector perpendicular to both input vectors. Its magnitude equals |a||b|sin(θ), where θ is the angle between the vectors. The direction follows the right-hand rule.

How do I calculate the cross product using determinants?

For vectors a = (ax, ay, az) and b = (bx, by, bz), set up a 3×3 determinant with unit vectors i, j, k in the first row, components of a in the second row, and components of b in the third row. Expand using cofactors to get: (ay*bz - az*by)i - (ax*bz - az*bx)j + (ax*by - ay*bx)k.

What is the right-hand rule for cross products?

The right-hand rule determines the direction of the cross product: Point your fingers in the direction of vector a, curl them toward vector b, and your thumb points in the direction of a × b. This is why a × b = -(b × a).

What does the magnitude of the cross product represent?

The magnitude |a × b| equals the area of the parallelogram formed by vectors a and b. When vectors are parallel (θ = 0° or 180°), the cross product is the zero vector because sin(0) = sin(180°) = 0.

What are the main properties of the cross product?

Key properties include: Anti-commutative (a × b = -b × a), Distributive (a × (b + c) = a × b + a × c), Scalar multiplication ((ka) × b = k(a × b)), and Zero product (a × a = 0).

What are real-world applications of cross products?

Cross products are used in physics for torque (τ = r × F), Lorentz force (F = qv × B), angular momentum (L = r × p), and in 3D computer graphics for calculating surface normals, lighting, and collision detection.

How is the cross product different from the dot product?

The cross product returns a vector and involves sin(θ), while the dot product returns a scalar and involves cos(θ). Cross product is anti-commutative and only exists in 3D (and 7D), while dot product is commutative and works in any dimension.

When is the cross product equal to zero?

The cross product a × b = 0 when either vector is the zero vector, or when the vectors are parallel (same or opposite direction). Mathematically, when a = kb for some scalar k, because sin(0°) = sin(180°) = 0.

Vector Algebra

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Vector Cross Product Knowledge

What is the Cross Product?

The cross product of two 3D vectors results in a third vector that is perpendicular to both. Unlike the dot product (which yields a scalar), the cross product captures the rotational and area-based relationship between vectors.

Geometric Meaning

The magnitude of the cross product equals the area of the parallelogram spanned by the two vectors. Its direction follows the Right-Hand Rule.

Frequently Asked Questions

The cross product is calculated using the determinant of a 3x3 matrix with unit vectors i, j, k in the first row, and the components of vectors A and B in the following rows.

Dot Product Calculator

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Vector Magnitude Calculator

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