Vector Calculator with Steps
This vector calculator combines two vectors in the plane or in space: enter the components and see the sum, difference, dot product, cross product, length, angle and projection at once. Pick a single operation and the tool shows the working with your numbers, step by step.
Source: Wikipedia – Dot product (Skalarprodukt, Winkelformel). Updated: .
How it is calculated
A vector describes a displacement with direction and length and is given by its components, in space by three numbers (x, y, z). For a = (1, 2, 3) and b = (4, 5, 6), for example, a + b = (5, 7, 9) and a · b = 1·4 + 2·5 + 3·6 = 32. The tool does this for any numbers, including decimals and negative values.
The operations at a glance
- Sum and difference: add or subtract component by component. The multiple k · a stretches or shrinks the vector; a negative k reverses its direction.
- Dot product: a · b = a₁b₁ + a₂b₂ + a₃b₃, a single number. If it is 0, the vectors are perpendicular (orthogonal).
- Cross product: in space, a × b is a vector perpendicular to both a and b. Its magnitude is the area of the parallelogram they span. In the plane the result is a single number, the z-component.
- Magnitude and angle: |a| = √(a₁² + a₂² + a₃²). The angle follows from cos φ = (a · b) ÷ (|a| · |b|) and lies between 0° and 180°. For whole numbers the root is also shown in exact form, such as √14.
- Projection: the part of a pointing along b is (a · b) ÷ (b · b) · b.
Which operation when?
Use the dot product to test orthogonality and compute angles, and the cross product for areas and normal vectors, for instance of planes. Parallel vectors show up as an angle of 0° or 180°. A line through two points can be set up with the equation of a line calculator, and matrices and determinants are handled by the matrix calculator. The formulas come from analytic geometry; results are shown to four decimal places.
Frequently asked questions
How do you calculate the dot product of two vectors?
Multiply the matching components and add the products: (1, 2, 3) · (4, 5, 6) = 4 + 10 + 18 = 32. The result is a number, not a vector.
How do you calculate the cross product?
In space with the formula a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). The result is perpendicular to both vectors; swapping the order flips the sign.
How do you find the angle between two vectors?
With cos φ = (a · b) ÷ (|a| · |b|), then φ = arccos(…). The tool shows the result in degrees and in radians.
How can I tell whether vectors are orthogonal or parallel?
They are orthogonal if the dot product is 0. They are parallel if the angle is 0° or 180°, that is, if one vector is a multiple of the other. The overview names the relationship.
How long is a vector?
Its length is the magnitude, the square root of the sum of the squared components. For (3, 4) that is √(9 + 16) = 5.
Sources and legal basis
- Wikipedia – Dot product (Skalarprodukt, Winkelformel)
- Wikipedia – Cross product (cross product, area of the parallelogram)
- Wikipedia – Vector projection (projection of a onto b)
As of: · Editorial policy
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