Vector Geometry Calculator

Work out vector geometry instantly with clear inputs, formula shown and shareable results.

About Vector Geometry Calculator

The dot product a·b = axbx + ayby + azbz measures projection and gives the angle through cos θ = a·b/(|a||b|). The cross product is perpendicular to both, and its magnitude equals the area of the parallelogram they span.

Formulas

Dot product

a · b = axbx + ayby + azbz = |a||b|cos θ

Cross product

a × b = (aybz − azby, azbx − axbz, axby − aybx)

Worked example

With the default values below, Vector Geometry Calculator returns Dot product a · b of 12.0000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Vector a (x, y, z): 1, 2, 3
Vector b (x, y, z): 4, -5, 6

Results

Dot product a · b: 12.0000
Cross product a × b: (27, 6, -13)
|a × b| (parallelogram area): 30.5614
Angle between the vectors: 68.5624 °

Step by step

1Values used
2Dot product
3Cross product
4Dot product a · b= 12.0000
5Cross product a × b= (27, 6, -13)
6|a × b| (parallelogram area)= 30.5614
7Angle between the vectors= 68.5624 °

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For a = (1,2,3), b = (4,−5,6)?

a·b = 4 − 10 + 18 = 12, and a × b = (27, 6, −13).

What does a zero cross product mean?

The vectors are parallel or antiparallel, so they span no area.