Cross Product Calculator
Free Cross product Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Formula
A x B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
The cross product is computed 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 second and third rows. The result is perpendicular to both input vectors.
Worked Examples
Example 1: Cross Product of Two 3D Vectors
Problem:Find the cross product of A = (2, 3, 4) and B = (5, 6, 7).
Solution:A x B = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1) = (3*7 - 4*6, 4*5 - 2*7, 2*6 - 3*5) = (21 - 24, 20 - 14, 12 - 15) = (-3, 6, -3) |A x B| = sqrt(9 + 36 + 9) = sqrt(54) = 7.3485 Parallelogram area = 7.3485 sq units
Result:Cross Product: (-3, 6, -3) | Magnitude: 7.3485 | Triangle Area: 3.6742
Example 2: Perpendicular Vectors Cross Product
Problem:Find A x B where A = (1, 0, 0) and B = (0, 1, 0).
Solution:A x B = (0*0 - 0*1, 0*0 - 1*0, 1*1 - 0*0) = (0, 0, 1) |A x B| = 1 Angle = 90 degrees (perpendicular vectors) The result (0, 0, 1) = k-hat, the unit z-vector
Result:Cross Product: (0, 0, 1) | Magnitude: 1 | Angle: 90 degrees
Frequently Asked Questions
What is the cross product of two vectors?
The cross product (also called the vector product) is a binary operation on two vectors in three-dimensional space that produces a third vector perpendicular to both input vectors. Unlike the dot product which yields a scalar, the cross product yields a vector. The direction of the resulting vector follows the right-hand rule: if you curl the fingers of your right hand from vector A toward vector B, your thumb points in the direction of A cross B. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors. The cross product is only defined for 3D vectors (and 7D, but that is rarely used).
How is the cross product calculated using the determinant formula?
The cross product A x B is calculated using a 3x3 determinant with the unit vectors i, j, k in the first row. For A = (a1, a2, a3) and B = (b1, b2, b3), the formula expands to: i(a2*b3 - a3*b2) - j(a1*b3 - a3*b1) + k(a1*b2 - a2*b1). This can be remembered by covering each column of the unit vectors and computing the 2x2 determinant of the remaining elements, alternating signs. The resulting vector (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1) is guaranteed to be perpendicular to both A and B, which can be verified by computing the dot product with each input vector.
What is the geometric meaning of the cross product magnitude?
The magnitude of the cross product |A x B| equals |A| * |B| * sin(theta), where theta is the angle between vectors A and B. Geometrically, this value represents the area of the parallelogram formed by the two vectors when placed tail-to-tail. Half of this magnitude gives the area of the triangle formed by the two vectors. When the vectors are parallel (theta = 0 or 180 degrees), the cross product magnitude is zero because the parallelogram degenerates into a line segment with no area. When the vectors are perpendicular (theta = 90 degrees), the magnitude is maximized at |A| * |B|. This area interpretation is widely used in physics and computer graphics.
What is the right-hand rule and how does it determine cross product direction?
The right-hand rule is a convention used to determine the direction of the cross product vector. To apply it, point the fingers of your right hand in the direction of the first vector A, then curl them toward the second vector B through the smaller angle between them. Your thumb will point in the direction of A x B. This means that the cross product is anti-commutative: A x B = -(B x A), so reversing the order reverses the direction. The right-hand rule is fundamental in physics for determining the direction of magnetic forces, torques, and angular momentum vectors. It establishes a consistent convention for defining positive rotation direction.
When is the cross product zero and what does it mean?
The cross product of two vectors is the zero vector when the input vectors are parallel or anti-parallel (pointing in the same or opposite directions). This occurs because the sine of 0 degrees and 180 degrees is zero, making |A x B| = |A|*|B|*sin(theta) = 0. The cross product is also zero if either input vector is the zero vector. Geometrically, parallel vectors cannot form a parallelogram with any area, so the cross product has zero magnitude. This property is commonly used as a test for parallelism: if A x B equals the zero vector, then A and B are parallel (or one is zero). In computational geometry, this test helps determine if line segments are collinear.
How is the cross product used in physics and engineering?
The cross product has extensive applications in physics and engineering. In electromagnetism, the magnetic force on a charged particle is F = qv x B (charge times velocity cross magnetic field). Torque is calculated as tau = r x F (position vector cross force). Angular momentum is L = r x p (position cross momentum). In fluid dynamics, the curl of a velocity field uses cross products to measure rotation. In structural engineering, moments about a point are computed using cross products. In computer graphics, cross products determine surface normals for lighting calculations and are used in ray-triangle intersection tests for 3D rendering algorithms.
What is the scalar triple product and how does it relate to the cross product?
The scalar triple product of three vectors A, B, and C is defined as A dot (B x C), which first computes the cross product of B and C, then takes the dot product with A. The result is a scalar whose absolute value equals the volume of the parallelepiped (3D parallelogram) formed by the three vectors. If the scalar triple product is positive, the vectors form a right-handed system; if negative, they form a left-handed system; if zero, the three vectors are coplanar. The scalar triple product can be computed directly as the determinant of a 3x3 matrix with the vectors as rows. This concept is essential in crystallography, continuum mechanics, and computational geometry.
Can the cross product be computed for vectors with more or fewer than three dimensions?
The traditional cross product is defined only for three-dimensional vectors. In two dimensions, a pseudo cross product can be defined that returns a scalar: for A = (a1, a2) and B = (b1, b2), the 2D cross product is a1*b2 - a2*b1, representing the signed area of the parallelogram and the z-component of the 3D cross product if z-values were zero. In seven dimensions, a cross product with similar properties exists but is rarely used in practice. For general n-dimensional spaces, the exterior product (wedge product) from exterior algebra generalizes the concept. The cross product does not satisfy the same algebraic properties in higher dimensions, which limits its extension.
What are the algebraic properties of the cross product?
The cross product has several important algebraic properties that distinguish it from other operations. It is anti-commutative: A x B = -(B x A). It distributes over addition: A x (B + C) = A x B + A x C. It is compatible with scalar multiplication: (kA) x B = k(A x B). However, it is NOT associative: A x (B x C) does not equal (A x B) x C in general. The BAC-CAB identity relates these: A x (B x C) = B(A dot C) - C(A dot B). The cross product of any vector with itself is always zero: A x A = 0. These properties are critical for simplifying expressions in physics and engineering calculations involving multiple cross products.
How is the unit normal vector obtained from the cross product?
The unit normal vector is obtained by normalizing the cross product vector, which means dividing it by its magnitude. If N = A x B, then the unit normal is N-hat = N / |N|. This unit vector has a magnitude of exactly 1 and points perpendicular to the plane containing vectors A and B. Unit normals are essential in computer graphics for determining how light reflects off surfaces (Phong shading, Lambertian reflectance). In computational geometry, unit normals define the orientation of planar surfaces. In physics, they define the direction perpendicular to a surface for computing flux integrals. The unit normal is undefined when A and B are parallel since the cross product is the zero vector.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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