What Does Cross Product Calculate In Math

Cross Product Formula:

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} \]

Vector A (x-component):

Vector A (y-component):

Vector A (z-component):

Vector B (x-component):

Vector B (y-component):

Vector B (z-component):

Cross Product (A × B):

Magnitude (Area of Parallelogram):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Does Cross Product Calculate?

The cross product calculates a vector that is perpendicular to both input vectors in 3D space. It also calculates the area of the parallelogram formed by the two vectors.

2. How Cross Product Works

The cross product of two vectors is calculated using the determinant formula:

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} \]

Which expands to:

\( x \)-component: \( A_yB_z - A_zB_y \)
\( y \)-component: \( A_zB_x - A_xB_z \)
\( z \)-component: \( A_xB_y - A_yB_x \)

3. Geometric Interpretation

Perpendicular Vector: The result is orthogonal to both input vectors, following the right-hand rule.

Magnitude: The length of the cross product equals the area of the parallelogram formed by the two vectors.

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The calculator will show the resulting vector and its magnitude.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between cross product and dot product?
A: Cross product gives a vector (with magnitude and direction), while dot product gives a scalar value representing projection.

Q2: Why is the cross product only defined in 3D?
A: The perpendicular vector concept only works consistently in three dimensions. In 2D, the result would need to be a scalar.

Q3: What does the magnitude represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q4: How is the direction determined?
A: By the right-hand rule - point fingers in direction of first vector, curl toward second vector, thumb points in cross product direction.

Q5: What if the cross product is zero?
A: A zero cross product means the vectors are parallel (or at least one is zero).