Cross Product Calculator UK

Cross Product Formula:

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

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 Result:

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1. What is the Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both original vectors, with magnitude equal to the area of the parallelogram that the vectors span.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

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

Where:

\( \mathbf{A} = (A_x, A_y, A_z) \) — First vector
\( \mathbf{B} = (B_x, B_y, B_z) \) — Second vector
\( \mathbf{i}, \mathbf{j}, \mathbf{k} \) — Unit vectors in x, y, z directions

Explanation: The result is calculated using the determinant of a 3×3 matrix with the unit vectors in the first row and the components of vectors A and B in the second and third rows.

3. Importance of Cross Product

Details: The cross product is essential in physics, engineering, and computer graphics for calculating torque, finding normal vectors to surfaces, and determining rotational effects.

4. Using the Calculator

Tips: Enter all six components (x, y, z for both vectors A and B). The calculator will compute the resulting vector that is perpendicular to both input vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar quantity, while cross product gives a vector quantity. Dot product measures parallelism, cross product measures perpendicularity.

Q2: What does the magnitude of the cross product represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q3: Can cross product be calculated in 2D?
A: The standard cross product is defined for 3D space. In 2D, you can compute a scalar value similar to the z-component of a 3D cross product.

Q4: What's the right-hand rule?
A: A mnemonic for determining the direction of the cross product vector. Point fingers in direction of first vector, curl toward second vector, thumb points in cross product direction.

Q5: When is the cross product zero?
A: When the vectors are parallel or at least one vector is zero.