Vector Calculator
Enter two vectors and get the sum, difference, dot and cross product, magnitudes, unit vector, angle and projection.
Vector calculator (2D and 3D)
Formulas
| Quantity | Formula |
|---|---|
| Magnitude | |A| = √(a₁² + a₂² + a₃²) |
| Dot product | A · B = a₁b₁ + a₂b₂ + a₃b₃ = |A||B| cos θ |
| Cross product (3D) | A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁) |
| Angle | θ = cos−1[(A · B)/(|A||B|)] |
| Unit vector | â = A/|A| |
| Projection of A on B | [(A · B)/|B|²] B |
Worked example
A = (1, 2, 3), B = (4, −1, 2). A · B = 4 − 2 + 6 = 8. A × B = (2·2 − 3·(−1), 3·4 − 1·2, 1·(−1) − 2·4) = (7, 10, −9). |A| = √14 ≈ 3.7417.
FAQ
How do I find the dot product?
Multiply matching components and add them. For (1,2,3) and (4,-1,2) it is 1*4 + 2*(-1) + 3*2 = 8.
What does the cross product give?
A vector perpendicular to both A and B, defined in 3D. Its length is the area of the parallelogram they span.
How do I enter a 2D vector?
Type two numbers such as 3, 4. For 2D vectors Calvo shows the scalar z-part of the cross product.
How is the angle found?
From the dot product: the angle is the inverse cosine of A dot B divided by the product of the magnitudes.
What if a vector is zero?
Angle, unit vector and projection are undefined for a zero vector, and Calvo tells you so.