Work out direction cosines instantly with clear inputs, formula shown and shareable results.
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.
Direction cosines
cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|
Identity
cos²α + cos²β + cos²γ = 1
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.
Because the direction cosines are exactly the components of the unit vector, whose length is 1.