Vector Calculator (3D Magnitude, Dot and Cross Product, Angle)
Enter the components of two vectors (use z = 0 for 2D).
Your numbers
Your results
|a|
|a|
—
|b|
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a · b (dot product)
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a × b (cross product)
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Angle between (degrees)
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Scalar projection of a on b
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Area of the parallelogram |a × b|
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Vector Calculator (3D Magnitude, Dot and Cross Product, Angle) by TheCalcTools
How the Vector Calculator (3D Magnitude, Dot and Cross Product, Angle) works
Magnitude |a| = √(x² + y² + z²). The dot product a · b = aₓbₓ + a_yb_y + a_zb_z equals |a||b|cos θ, which gives the angle.
The cross product is a vector perpendicular to both, with length |a||b|sin θ, the area of the parallelogram they span.
The formula
a · b = Σ aᵢbᵢ = |a||b| cos θa × b = (a_yb_z − a_zb_y, a_zbₓ − aₓb_z, aₓb_y − a_ybₓ)
Worked example
Using the default values shown in the calculator:
| a: x | 3 |
|---|---|
| a: y | -2 |
| a: z | 5 |
| b: x | 1 |
| b: y | 4 |
| b: z | -2 |
- a = (3, −2, 5), b = (1, 4, −2): a · b = -15.0000, a × b = (-16, 11, 14).
- The angle between them is 122.0727°.
Result:
| |a| | 6.1644 |
|---|---|
| |b| | 4.5826 |
| a · b (dot product) | -15.0000 |
| a × b (cross product) | (-16, 11, 14) |
| Angle between (degrees) | 122.0727 |
| Scalar projection of a on b | -3.2733 |
| Area of the parallelogram |a × b| | 23.9374 |
Frequently asked questions
When is the dot product zero?
When the vectors are perpendicular (or one is zero).
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.