Cross Product Calculator

What Is the Cross Product?

The cross product (also called the vector product) is an operation on two vectors in three-dimensional space that produces a third vector perpendicular to both inputs. If vector A = [a₁, a₂, a₃] and vector B = [b₁, b₂, b₃], the cross product A x B = [a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁]. The result vector is perpendicular to the plane containing A and B, and its magnitude equals the area of the parallelogram formed by A and B. Enter two vectors in the calculator above to compute the cross product with detailed steps.

How to Calculate the Cross Product?

The most common method uses the determinant formula with unit vectors i, j, k. Write a 3x3 matrix with i, j, k in the first row, vector A components in the second row, and vector B components in the third row. Expand the determinant along the first row. For A = [2, 3, 4] and B = [5, 6, 7]: A x B = i(3*7 - 4*6) - j(2*7 - 4*5) + k(2*6 - 3*5) = i(21-24) - j(14-20) + k(12-15) = [-3, 6, -3]. The memory trick is to cover one column at a time and compute the 2x2 determinant of the remaining entries, alternating signs (+, -, +).

Properties of the Cross Product

The cross product is anticommutative: A x B = -(B x A). Reversing the order flips the direction of the result. The cross product of parallel vectors is the zero vector because the parallelogram they form has zero area. The cross product of a vector with itself is always zero. The magnitude of A x B equals |A||B|sin(theta), where theta is the angle between them. Maximum magnitude occurs when vectors are perpendicular (sin 90 = 1). The cross product distributes over addition: A x (B + C) = A x B + A x C. Unlike the dot product (which produces a scalar), the cross product produces a vector.

Cross Product vs Dot Product

The dot product A dot B = a₁b₁ + a₂b₂ + a₃b₃ produces a scalar and measures how much two vectors point in the same direction. The cross product A x B produces a vector perpendicular to both and measures the area of the parallelogram they form. The dot product uses cosine of the angle between vectors (maximum when parallel). The cross product uses sine (maximum when perpendicular). Both operations are fundamental to vector algebra, but they serve different purposes: dot products measure projection and work, while cross products measure rotation and area.

Applications of the Cross Product

Physics uses the cross product extensively. Torque is the cross product of the position vector and force: tau = r x F. The magnetic force on a moving charge is F = qv x B (charge times velocity cross magnetic field). Angular momentum is L = r x p (position cross momentum). In computer graphics, the cross product finds surface normals for lighting calculations. Given two edges of a polygon, their cross product gives the perpendicular direction used to determine how light reflects off the surface. In navigation, the cross product of two position vectors determines if a turn is clockwise or counterclockwise.

Finding the Area of a Triangle or Parallelogram

The magnitude of the cross product of two vectors equals the area of the parallelogram they span. The area of the triangle formed by the same vectors is half that: Area = |A x B| / 2. For vectors A = [3, 0, 0] and B = [0, 4, 0]: A x B = [0, 0, 12], magnitude = 12. Parallelogram area = 12, triangle area = 6. This method works in 3D space where the standard base-height formula is difficult to apply because the height may not align with any axis. Given three points in space, compute two edge vectors and take half the cross product magnitude for the triangle area.

The Right-Hand Rule

The right-hand rule determines the direction of the cross product vector. Point your right hand fingers in the direction of the first vector A. Curl your fingers toward the second vector B. Your thumb points in the direction of A x B. This convention ensures the cross product is consistently defined and is why A x B and B x A point in opposite directions. The right-hand rule is used throughout physics for determining the direction of magnetic fields, torques, and angular velocities. In coordinate systems, i x j = k, j x k = i, and k x i = j all follow the right-hand rule.

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