Cross Product Calculator

The Cross Product Calculator helps you find the cross product of two vectors in three-dimensional space. The cross product is commonly used in mathematics, physics, engineering, and geometry to calculate a vector that is perpendicular to two given vectors.

What Is a Cross Product?

The cross product, also called the vector product, combines two three-dimensional vectors to produce another vector. If the vectors are A and B, their cross product is written as:

A × B

The resulting vector is perpendicular to both A and B.

Cross Product Formula

For two vectors:

A = (A_x, A_y, A_z)

B = (B_x, B_y, B_z)

The cross product is:

A × B = (A_yB_z − A_zB_y, A_zB_x − A_xB_z, A_xB_y − A_yB_x)

How to Use the Cross Product Calculator

1. Enter the three components of the first vector.
2. Enter the three components of the second vector.
3. Click the calculate button.
4. The calculator will return the resulting cross product vector.

Cross Product Example

Suppose we have:

A = (1, 2, 3)

B = (4, 5, 6)

Using the cross product formula:

A × B = (2 × 6 − 3 × 5, 3 × 4 − 1 × 6, 1 × 5 − 2 × 4)

Therefore:

A × B = (-3, 6, -3)

Properties of the Cross Product

The cross product has several important properties:

• Anti-commutative: A × B = −(B × A)
• Perpendicular: The resulting vector is perpendicular to both input vectors.
• Magnitude: |A × B| = |A||B|sin(θ)
• Parallel vectors: If two vectors are parallel, their cross product is zero.

Cross Product and the Angle Between Vectors

The magnitude of a cross product can be calculated using the angle between the two vectors:

|A × B| = |A||B|sin(θ)

Here, θ is the angle between the vectors. The magnitude is greatest when the vectors are perpendicular and is zero when they are parallel.

Applications of the Cross Product

Cross products have practical applications in many areas, including:

• Physics and mechanics
• Engineering
• Three-dimensional geometry
• Computer graphics
• Robotics
• Calculating torque and angular momentum

Frequently Asked Questions

What does a cross product calculate?

A cross product produces a vector that is perpendicular to the two original vectors.

Can the cross product be used with two-dimensional vectors?

The standard cross product is defined for three-dimensional vectors. Two-dimensional vectors can be extended into three dimensions by assigning a zero z-component when appropriate.

What happens when two vectors are parallel?

Their cross product is the zero vector because the sine of the angle between parallel vectors is zero.

Related Calculators

If you are working with vectors and mathematical operations, you can also explore our Math Calculators for additional tools and calculations.

Conclusion

The Cross Product Calculator provides a quick way to calculate the vector product of two three-dimensional vectors. It can save time when solving geometry, physics, engineering, and other vector-based problems while helping you verify your calculations.