Cross Product Calculator – Calculate Vector Cross Products

Cross Product Calculator

Easily calculate the cross product of two 3D vectors. Input the components of Vector A and Vector B to find their cross product Vector C.

Calculate Cross Product

Enter the components of the two vectors below:

Vector A (a):

a_x (i component)

a_y (j component)

a_z (k component)

Vector B (b):

b_x (i component)

b_y (j component)

b_z (k component)

Result (a x b):

( -3 , 6 , -3 )

Intermediate Calculations:

The cross product a x b = (a_yb_z – a_zb_y)i + (a_zb_x – a_xb_z)j + (a_xb_y – a_yb_x)k

Vector Components Visualization

Vector A Vector B A x B

Bar chart showing x, y, and z components of each vector.

What is the Cross Product?

The Cross Product Calculator helps you find the cross product (or vector product) of two vectors in three-dimensional space. The cross product of two linearly independent vectors, say vector A and vector B, is a vector C that is perpendicular to both A and B. Its direction is given by the right-hand rule, and its magnitude is equal to the area of the parallelogram that the vectors A and B span.

This calculator is useful for students, engineers, physicists, and anyone working with 3D geometry or vector algebra. It's commonly used in physics to calculate torque, angular momentum, or the force on a moving charge in a magnetic field. Our Cross Product Calculator provides quick and accurate results.

A common misconception is that the cross product is commutative (A x B = B x A). However, it is anti-commutative, meaning A x B = – (B x A). The Cross Product Calculator respects this property.

Cross Product Formula and Mathematical Explanation

Given two vectors A = (a_x, a_y, a_z) and B = (b_x, b_y, b_z), their cross product A x B is defined as:

A x B = (a_yb_z – a_zb_y)i + (a_zb_x – a_xb_z)j + (a_xb_y – a_yb_x)k

Where i, j, and k are the standard unit vectors along the x, y, and z axes, respectively. This can also be represented as the determinant of a matrix:

A x B = | i j k |
| a_x a_y a_z |
| b_x b_y b_z |

Expanding the determinant gives:

x-component (i): (a_y * b_z) – (a_z * b_y)
y-component (j): (a_z * b_x) – (a_x * b_z)
z-component (k): (a_x * b_y) – (a_y * b_x)

Our Cross Product Calculator performs these calculations for you.

Variables Table

Variable	Meaning	Unit	Typical Range
a_x, a_y, a_z	Components of Vector A	Dimensionless (or units of the vector quantity)	Any real number
b_x, b_y, b_z	Components of Vector B	Dimensionless (or units of the vector quantity)	Any real number
c_x, c_y, c_z	Components of the Resultant Cross Product Vector C (A x B)	Product of units of A and B	Any real number

Table 1: Variables used in the Cross Product Calculator.

Practical Examples (Real-World Use Cases)

Example 1: Finding a Normal Vector

Suppose you have two vectors in a plane, A = (2, 1, 0) and B = (1, 3, 0). Their cross product will give a vector perpendicular to the plane containing A and B.

Using the Cross Product Calculator with A = (2, 1, 0) and B = (1, 3, 0):

A x B = ((1*0) – (0*3))i + ((0*1) – (2*0))j + ((2*3) – (1*1))k = 0i + 0j + 5k = (0, 0, 5).

The resulting vector (0, 0, 5) is along the z-axis, perpendicular to the xy-plane where A and B lie.

Example 2: Calculating Torque

Torque (τ) can be calculated as the cross product of the position vector (r) from the axis of rotation to the point of force application, and the force vector (F): τ = r x F.

If r = (1, 1, 0) meters and F = (0, 10, 0) Newtons:

Using the Cross Product Calculator with r = (1, 1, 0) and F = (0, 10, 0):

τ = ((1*0) – (0*10))i + ((0*0) – (1*0))j + ((1*10) – (1*0))k = 0i + 0j + 10k = (0, 0, 10) Newton-meters.

The torque is 10 Nm along the z-axis.

How to Use This Cross Product Calculator

Enter Vector A Components: Input the x (a_x), y (a_y), and z (a_z) components of the first vector into the respective fields under "Vector A".
Enter Vector B Components: Input the x (b_x), y (b_y), and z (b_z) components of the second vector into the respective fields under "Vector B".
View Results: The Cross Product Calculator automatically computes and displays the components of the resultant vector (A x B) in the "Result" section in real-time. Intermediate calculation steps are also shown.
See Visualization: The bar chart visualizes the x, y, and z components of Vector A, Vector B, and the resultant cross product vector.
Reset: Click the "Reset" button to clear the input fields and restore default values.
Copy Results: Click "Copy Results" to copy the main result and intermediate values to your clipboard.

The result is a vector perpendicular to both input vectors. Its magnitude represents the area of the parallelogram formed by the two vectors. Our vector calculator can help with other vector operations.

Key Factors That Affect Cross Product Results

The result of the cross product A x B is directly influenced by:

Components of Vector A: Changing any component of vector A (a_x, a_y, a_z) will alter both the direction and magnitude of the resultant vector, as each component of the result depends on the components of A.
Components of Vector B: Similarly, changes to b_x, b_y, or b_z will affect the cross product.
Order of Vectors: The cross product is anti-commutative (A x B = – B x A). Swapping the order of the vectors will negate the resultant vector, reversing its direction but keeping the same magnitude. The Cross Product Calculator computes A x B.
Angle Between Vectors: The magnitude of A x B is |A||B|sin(θ), where θ is the angle between A and B. If the vectors are parallel (θ=0°) or anti-parallel (θ=180°), sin(θ)=0, and the cross product is the zero vector (0, 0, 0). If they are perpendicular (θ=90°), sin(θ)=1, and the magnitude is maximized.
Magnitude of Input Vectors: Increasing the magnitude of either vector A or B (while keeping the angle constant and non-zero) will increase the magnitude of the cross product proportionally.
Linear Dependence: If the vectors are linearly dependent (one is a scalar multiple of the other, meaning they are parallel or anti-parallel), their cross product is zero. The Cross Product Calculator will show (0, 0, 0).

Frequently Asked Questions (FAQ)

What is the cross product of two parallel vectors?: The cross product of two parallel or anti-parallel vectors is the zero vector (0, 0, 0) because the angle between them is 0 or 180 degrees, and sin(0) = sin(180) = 0. Our Cross Product Calculator will yield (0,0,0).
Is the cross product defined for 2D vectors?: Strictly speaking, the cross product as a vector is defined only for 3D vectors. However, if you consider 2D vectors (a_x, a_y) and (b_x, b_y) as 3D vectors with z-components equal to zero, (a_x, a_y, 0) and (b_x, b_y, 0), their cross product is (0, 0, a_xb_y – a_yb_x), a vector along the z-axis. The magnitude a_xb_y – a_yb_x is sometimes referred to as the 2D cross product magnitude.
What is the geometric meaning of the cross product's magnitude?: The magnitude of the cross product A x B is equal to the area of the parallelogram formed by vectors A and B as adjacent sides.
What is the right-hand rule?: The right-hand rule determines the direction of A x B. If you point your right index finger along A and your middle finger along B, your thumb will point in the direction of A x B.
Can I use the Cross Product Calculator for vectors with negative components?: Yes, the Cross Product Calculator accepts positive, negative, and zero values for the vector components.
How is the cross product different from the dot product?: The cross product of two vectors results in a vector perpendicular to both, while the dot product results in a scalar (a single number). You can use our dot product calculator for that.
What happens if I enter non-numeric values?: The Cross Product Calculator will show an error message and will not perform the calculation until valid numbers are entered.
Does the order of vectors matter in the cross product?: Yes, A x B = -(B x A). Reversing the order changes the direction of the resulting vector. Our Cross Product Calculator calculates A x B.

Related Tools and Internal Resources

Vector Calculator: Perform various vector operations like addition, subtraction, and scalar multiplication.
Dot Product Calculator: Calculate the dot product of two vectors.
Matrix Calculator: Useful for operations involving matrices, including determinants related to cross products.
Physics Calculators: Explore other calculators related to physics concepts.
3D Vector Tool: More tools for working with three-dimensional vectors.
Vector Algebra Basics: Learn more about the fundamentals of vector algebra.

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