The dot and cross products are the two ways to "multiply" vectors, and they answer completely different questions. A vector calculator gives you both instantly, but knowing what each result means is what makes them useful in physics and geometry.
Throughout we use A = (1, 2, 3) and B = (4, 5, 6).
On this page
Magnitude
The magnitude |A| is the vector's length, from the square root of the sum of squared components.
|A| = √(1² + 2² + 3²) = √14 ≈ 3.742.
The dot product
The dot product multiplies matching components and adds them, giving a single number (a scalar):
A · B = a₁b₁ + a₂b₂ + a₃b₃A · B = 1·4 + 2·5 + 3·6 = 4 + 10 + 18 = 32.
Geometrically A · B = |A||B|cos θ, so it measures how aligned the vectors are. It's zero exactly when they're perpendicular. It powers work (force · displacement) and projections.
The cross product
The cross product returns a vector perpendicular to both inputs (3D only):
A × B = ( a₂b₃ − a₃b₂ , a₃b₁ − a₁b₃ , a₁b₂ − a₂b₁ )A × B = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (12 − 15, 12 − 6, 5 − 8) = (−3, 6, −3).
Its length |A × B| = |A||B|sin θ equals the area of the parallelogram the two vectors span, and it's zero when they're parallel. The direction follows the right-hand rule. It powers torque, normals and rotational motion.
Which one do I need?
| Dot product | Cross product | |
|---|---|---|
| Result | Scalar (number) | Vector |
| Zero when | Perpendicular | Parallel |
| Measures | Alignment / projection | Perpendicular direction / area |
| Typical use | Work, angle between vectors | Torque, surface normal |
Computing it online
Switch to VECTOR mode. Enter Vector A's components (spaces or commas), then Vector B. The calculator returns |A|, |B|, A · B (dot), A × B (cross) and A + B together — reproducing every result above. Vectors and matrices share the same linear-algebra toolkit; see the matrix calculator pillar guide for the matrix side.
Frequently asked questions
What's the difference between dot and cross?
Dot returns a scalar measuring alignment; cross returns a perpendicular vector whose length is the spanned parallelogram's area.
How do I compute a dot product?
Multiply matching components and add: (1,2,3)·(4,5,6) = 32.
When is the cross product zero?
When the vectors are parallel; the dot product is zero when they're perpendicular.
Compute dot & cross instantly
Enter two vectors in VECTOR mode and read magnitude, dot and cross at once.
Open the vector calculator →