Vector calculator
Compute dot product, magnitudes, and the 3D cross product for two vectors.
What this calculator covers
Use this vector calculator to compare two vectors in either 2D or 3D space.
It keeps the scalar and vector summaries together, which makes it useful for coordinate geometry, basic physics, and linear algebra review.
Frequently asked questions
- What does the dot product tell you?
- The dot product is a scalar that measures how much two vectors point in the same direction. A positive result means the vectors have a component in the same direction, zero means they are perpendicular, and a negative result means they point in generally opposite directions.
- Why is the cross product only available in 3D mode?
- The standard vector cross product is defined for three-dimensional vectors and produces a new vector perpendicular to both inputs. There is no direct equivalent for 2D vectors, so the calculator only reports the cross product when 3D is selected.
- How is vector magnitude calculated?
- Magnitude is the length of a vector, computed as the square root of the sum of its squared components. For a vector (a, b, c), the magnitude is √(a² + b² + c²).
- Can I use this for unit vector calculations?
- Not directly — the calculator reports raw magnitudes but does not normalize vectors. To find a unit vector, divide each component by the magnitude shown in the output.
Tool
Run the calculation
Result
RESULT · VECTOR
â„–219
Primary result
dot = 32; |a| = 3.741657; |b| = 8.774964; a x b = (-3, 6, -3)
For vectors a = (1, 2, 3) and b = (4, 5, 6), the dot product is 32, |a| = 3.741657, and |b| = 8.774964, with a x b = (-3, 6, -3).
- Vector a
- (1, 2, 3)
- Vector b
- (4, 5, 6)
- Dot product
- 32
- Magnitudes
- |a| = 3.741657, |b| = 8.774964
- Cross product
- (-3, 6, -3)
Step-by-step solution
- 1.Multiply matching components and add them to get the dot product 32.
- 2.Square each vector component, sum the squares, and take square roots to get |a| = 3.741657 and |b| = 8.774964.
- 3.Use the 3D cross-product determinant pattern to get a x b = (-3, 6, -3).
Walkthrough
Visual walkthrough
Vector calculations compare direction and size component by component, then combine the results into scalar or vector summaries.
01
Lay out the input vectors
a = (1, 2, 3), b = (4, 5, 6)
The vectors are compared component by component in the selected dimension.
02
Compute the scalar summaries
a · b = 32, |a| = 3.741657, |b| = 8.774964
The dot product measures directional agreement, while the magnitudes measure each vector's length.
03
Compute the 3D cross product
In 3D, the cross product produces a new vector perpendicular to both inputs.
(-3, 6, -3)