Direction Cosines Calculator

Work out direction cosines instantly with clear inputs, formula shown and shareable results.

About Direction Cosines Calculator

Direction cosines are the cosines of the angles a vector makes with each coordinate axis, obtained by dividing each component by the magnitude. They always satisfy cos²α + cos²β + cos²γ = 1.

Formulas

Direction cosines

cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|

Identity

cos²α + cos²β + cos²γ = 1

Worked example

With the default values below, Direction Cosines Calculator returns cos α (with the x-axis) of 0.33333333. Change any input above and every figure here is recalculated in your browser.

Inputs used

Vector (x, y, z): 1, 2, 2
Scale the vector by: 1

Results

cos α (with the x-axis): 0.33333333
cos β (with the y-axis): 0.66666667
cos γ (with the z-axis): 0.66666667
cos²α + cos²β + cos²γ (must be 1): 1.0000000000

Step by step

1Values used
2Direction cosines
3Identity
4cos α (with the x-axis)= 0.33333333
5cos β (with the y-axis)= 0.66666667
6cos γ (with the z-axis)= 0.66666667
7cos²α + cos²β + cos²γ (must be 1)= 1.0000000000

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 10 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 (1, 2, 2)?

The magnitude is 3, so the cosines are 1/3, 2/3 and 2/3, and their squares sum to 1/9 + 4/9 + 4/9 = 1.

Why must the squares sum to 1?

Because the direction cosines are exactly the components of the unit vector, whose length is 1.