Cross Product of 2 3D Vectors Calculator With Points

Cross Product Formula:

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \\ \end{vmatrix} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} \]

Vector A (P2 - P1)

Point 1 (P1) X:

Point 1 (P1) Y:

Point 1 (P1) Z:

Point 2 (P2) X:

Point 2 (P2) Y:

Point 2 (P2) Z:

Vector B (P4 - P3)

Point 3 (P3) X:

Point 3 (P3) Y:

Point 3 (P3) Z:

Point 4 (P4) X:

Point 4 (P4) Y:

Point 4 (P4) Z:

Vector A (P2-P1):

Vector B (P4-P3):

Cross Product (A × B):

Magnitude of Cross Product:

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1. What is the Cross Product of Two Vectors?

The cross product is a binary operation on two vectors in three-dimensional space that results in a vector perpendicular to both original vectors. Its magnitude equals the area of the parallelogram formed by the two vectors.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

\[ \vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} \]

Where:

\( \vec{A} = (A_x, A_y, A_z) \) — First vector (P2 - P1)
\( \vec{B} = (B_x, B_y, B_z) \) — Second vector (P4 - P3)
\( \mathbf{i}, \mathbf{j}, \mathbf{k} \) — Unit vectors in x, y, z directions

Explanation: The calculator first computes the vectors from the given points (P2-P1 and P4-P3), then applies the cross product formula.

3. Importance of Cross Product

Details: Cross products are essential in physics (torque, angular momentum), computer graphics (surface normals), and engineering (moment of forces).

4. Using the Calculator

Tips: Enter coordinates for four points (P1, P2 for vector A; P3, P4 for vector B). The calculator will compute vectors A and B from these points, then their cross product.

5. Frequently Asked Questions (FAQ)

Q1: What does the cross product magnitude represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q2: When is the cross product zero?
A: When vectors are parallel or one/both are zero vectors.

Q3: What's the right-hand rule?
A: Point fingers in direction of first vector, curl toward second vector; thumb points in cross product direction.

Q4: Can this be used for 2D vectors?
A: Yes, treat them as 3D with z=0. The result will be along the z-axis.

Q5: What's the difference between dot and cross products?
A: Dot product gives a scalar (number), cross product gives a vector perpendicular to both inputs.