Vector Dot & Cross Product Calculator
Calculate the dot product, cross product, magnitudes, and angle between two 3D vectors. Free vector mathematics and scalar calculator.
Interactive Calculator
1. Provide Details
2. Output Results
How to Calculate: Formula & Steps
Computes the scalar dot product by summing the products of corresponding components, and computes the 3D cross product vector perpendicular to both input vectors.
A · B = AxBx + AyBy + AzBz; A × B = [AyBz - AzBy, AzBx - AxBz, AxBy - AyBx]Step-by-Step Calculation Example
For A = [3, -5, 4] and B = [2, 7, -3]: Dot Product is -41; Cross Product is [-13, 17, 31].
Common Mistakes to Avoid
- Confusing the dot product (which outputs a single number) with the cross product (which outputs a new 3D vector).
- Attempting to calculate a cross product for 2D vectors directly (the cross product is only defined in three dimensions, though 2D vectors can be represented in 3D by setting the Z component to 0).
Practical Use Cases
- Physics force and torque calculations
- 3D computer graphics and game development
- Linear algebra homework checking
- Engineering structural analysis
Expert Tips
- To calculate 2D vectors, simply leave the Z component (Az and Bz) empty or set to 0.
- The order of vectors matters for the cross product: A × B is the negative (opposite direction) of B × A.
Frequently Asked Questions About Vector Dot & Cross Product Calculator
What is a dot product (scalar product)?
The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single scalar number. It measures how much the vectors point in the same direction.
What is a cross product (vector product)?
The cross product is a binary operation on two vectors in a three-dimensional space. It results in a third vector that is perpendicular (orthogonal) to both of the original vectors, with a magnitude equal to the area of the parallelogram that the vectors span.
How do you calculate the dot product of two vectors?
Multiply the corresponding components of the two vectors together and sum the results: A · B = (Ax × Bx) + (Ay × By) + (Az × Bz).
How do you calculate the cross product of two vectors?
Use the determinant formula for 3D vectors: AxB = [ (Ay × Bz - Az × By), (Az × Bx - Ax × Bz), (Ax × By - Ay × Bx) ]. The result is always a 3D vector, not a scalar.
What is the angle between perpendicular and parallel vectors?
If the dot product of two vectors is exactly 0, the vectors are perpendicular (90 degrees). If the cross product of two vectors is [0, 0, 0], the vectors are parallel (0 or 180 degrees).
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