What is a cross product in mathematics?

In mathematics, a cross product is a method of multiplying two vectors that results in a third vector. This resulting vector is perpendicular to both of the original vectors. The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors, calculated as the product of their magnitudes and the sine of the angle between them. The direction of the resulting vector is determined by the right-hand rule . Unlike the dot product, which yields a scalar, the cross product produces a vector.

Which kinds of mathematical objects use the cross product?

The cross product is a mathematical operation used with vectors in three-dimensional space. It produces a new vector that is perpendicular to both of the original vectors. This operation is fundamental in various scientific applications. Key applications include calculating torque , which describes the rotational force applied to an object, and the Lorentz force , which explains the force exerted on a charged particle moving through electric and magnetic fields. The magnitude of the cross product also represents the area of the parallelogram formed by the two vectors.

How is the cross product different from the dot product?

The cross product and dot product are distinct methods of multiplying vectors. The dot product , also known as the scalar product, results in a scalar (a single number). This scalar is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. In contrast, the cross product produces a new vector that is perpendicular to both of the original vectors. The magnitude of this resulting vector is equal to the area of the parallelogram formed by the two input vectors. Its direction is determined by the right-hand rule. The cross product is specifically defined in three-dimensional space.

How do you find the cross product of two vectors in 3D?

To find the cross product of two vectors in 3D, let the vectors be (a = (a_x, a_y, a_z)) and (b = (b_x, b_y, b_z)). The cross product, denoted as (a \times b), results in a vector perpendicular to both (a) and (b). The formula for the cross product is: (a \times b = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_x & a_y & a_z \ b_x & b_y & b_z \end{vmatrix}) This determinant expands to: (a \times b = \mathbf{i}(a_y b_z - a_z b_y) - \mathbf{j}(a_x b_z - a_z b_x) + \mathbf{k}(a_x b_y - a_y b_x)) The direction of the resulting vector is determined by the right-hand rule.

What does the direction of the cross product represent?

The direction of the cross product of two vectors, denoted as a × b , represents a vector that is perpendicular to both a and b . This direction is determined by the right-hand rule . If you place the heel of your right hand at the point where the tails of the vectors meet and curl your fingers from vector a to vector b , your thumb will point in the direction of the cross product c . This concept is fundamental in three-dimensional space, where the resulting vector will be parallel to the z-axis if the initial vectors lie in the xy-plane. The cross product is also known as the vector product .

How is the cross product used in real-world applications?

The cross product, a mathematical operation yielding a vector perpendicular to two input vectors, finds practical use in several real-world applications. It is fundamental in calculating torque , which describes the rotational effect of a force, essential for understanding how screws are installed or how bicycle pedals propel a bike. The equation for torque is (\tau = F \times r), where (\tau) is torque, (F) is force, and (r) is the vector from the rotational axis to the point of force application. Additionally, the cross product is crucial for determining the Lorentz force , which is the force experienced by a charged particle moving through electric and magnetic fields. This concept is vital in understanding electromagnetism and its applications.

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cross product

mathematics

Also known as: vector product

Written by

Ken Stewart

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Last updated

Oct. 21, 2026 •History

Top Questions

What is a cross product in mathematics?
In mathematics, a cross product is a method of multiplying two vectors that results in a third vector. This resulting vector is perpendicular to both of the original vectors. The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors, calculated as the product of their magnitudes and the sine of the angle between them. The direction of the resulting vector is determined by the right-hand rule. Unlike the dot product, which yields a scalar, the cross product produces a vector.
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Which kinds of mathematical objects use the cross product?
The cross product is a mathematical operation used with vectors in three-dimensional space. It produces a new vector that is perpendicular to both of the original vectors. This operation is fundamental in various scientific applications.

Key applications include calculating torque, which describes the rotational force applied to an object, and the Lorentz force, which explains the force exerted on a charged particle moving through electric and magnetic fields. The magnitude of the cross product also represents the area of the parallelogram formed by the two vectors.
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How is the cross product different from the dot product?
The cross product and dot product are distinct methods of multiplying vectors. The dot product, also known as the scalar product, results in a scalar (a single number). This scalar is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them.

In contrast, the cross product produces a new vector that is perpendicular to both of the original vectors. The magnitude of this resulting vector is equal to the area of the parallelogram formed by the two input vectors. Its direction is determined by the right-hand rule. The cross product is specifically defined in three-dimensional space.
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2. cross product
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How do you find the cross product of two vectors in 3D?
To find the cross product of two vectors in 3D, let the vectors be (a = (a_x, a_y, a_z)) and (b = (b_x, b_y, b_z)). The cross product, denoted as (a \times b), results in a vector perpendicular to both (a) and (b).

The formula for the cross product is:
(a \times b = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_x & a_y & a_z \ b_x & b_y & b_z \end{vmatrix})

This determinant expands to:
(a \times b = \mathbf{i}(a_y b_z - a_z b_y) - \mathbf{j}(a_x b_z - a_z b_x) + \mathbf{k}(a_x b_y - a_y b_x))

The direction of the resulting vector is determined by the right-hand rule.
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What does the direction of the cross product represent?
The direction of the cross product of two vectors, denoted as a × b, represents a vector that is perpendicular to both a and b. This direction is determined by the right-hand rule. If you place the heel of your right hand at the point where the tails of the vectors meet and curl your fingers from vector a to vector b, your thumb will point in the direction of the cross product c. This concept is fundamental in three-dimensional space, where the resulting vector will be parallel to the z-axis if the initial vectors lie in the xy-plane. The cross product is also known as the vector product.
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1. cross product
2. cross product
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How is the cross product used in real-world applications?
The cross product, a mathematical operation yielding a vector perpendicular to two input vectors, finds practical use in several real-world applications. It is fundamental in calculating torque, which describes the rotational effect of a force, essential for understanding how screws are installed or how bicycle pedals propel a bike. The equation for torque is (\tau = F \times r), where (\tau) is torque, (F) is force, and (r) is the vector from the rotational axis to the point of force application.

Additionally, the cross product is crucial for determining the Lorentz force, which is the force experienced by a charged particle moving through electric and magnetic fields. This concept is vital in understanding electromagnetism and its applications.
4 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

cross product, a method of multiplying two vectors that produces a vector perpendicular to both vectors involved in the multiplication; that is, a × b = c, where c is perpendicular to both a and b. The magnitude of c is given by the product of the magnitudes of a and b and the sine of the angle θ between a and b, that is, |a × b| = |c| = |a| |b| sin θ.Thus the magnitude of c is the area of the parallelogram formed by a and b, with |a| being the base and |b| sin θ being the height of the parallelogram. The cross product is distinguished from the dot product, which produces a scalar when multiplying two vectors.

The direction of c is found using the right-hand rule. This rule indicates that the heel of the right hand is placed at the point where the two tails of the vectors are connected, and the fingers of the right hand then wrap in a direction from a to b. When this is done, the thumb of the right hand will point in the direction of the cross product c. Clearly, from this definition, the vector space for a cross product is three-dimensional space. If, for example, the two given vectors in the cross product are both in the xy plane, the resulting vector is perpendicular to these two vectors, and this means a vector that is parallel to the z-axis.

For the two vectors a = (a_x, a_y, a_z) and b = (b_x, b_y, b_z), the cross product is found by calculating the determinant of the matrix with the unit vectors x, y, and z being the first row and the vectors a and b being the last two rows. The determinant creates the following formula for the cross product:a × b = x(a_yb_z − a_zb_y) + y(a_zb_x − a_xb_z) + z(a_xb_y − a_yb_x)

If a and b are parallel, a × b = 0. Also, since rotation from b to a is opposite to that from a to b,a × b = −b × a.This shows that the cross product is not commutative, but the distributive law a × (b + d) = (a × b) + (a × d)holds. Other properties include the Jacobi property, a × (b × c) + b × (c × a) + c × (a × b) = 0;the scalar multiple property, given a constant k,k(a × b) = ka × b = a × kb;and the zero vector property, a × b = 0, where either a or b is the zero vector, with all elements equal to zero.

The cross product has many applications in science. One such example is torque, which allows screws to be installed and allows a bicycle’s pedals to move it forward. The equation for torque is τ = F × r, where τ is torque, F is the applied force, and r is the vector from the rotational axis to the place where the force is applied.

Another prominent example is the Lorentz force, the force exerted on a charged particle q moving with velocity v through an electric field E and magnetic field B. The entire electromagnetic force F on the charged particle is given by F = qE + qv × B.

Ken Stewart

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