Cartesian components and direction cosines

Cartesian Components and Direction Cosines

Choose a right-handed orthonormal Cartesian basis

Every vector has the unique component form

For a point P = (x,y,z),

Hence

The magnitude is

If α,β,γ are the direction angles,

so

For planar vectors it is better numerically to use

rather than reconstructing the quadrant from tan 𝜃 = u_y∕u_x.

Source examples

For u = (−2, 1, 2),

with direction cosines (−2∕3, 1∕3, 2∕3).

If u = (5, 2) and v = (−3,−4), then

and 𝜃 = −45^∘ (equivalently 315^∘).

A line through P₁ = (x₁,y₁,z₁) parallel to ℓ = (a,b,c) can be written parametrically as

or, when the denominators are nonzero,

Source problems

1. Find u + v + w graphically and analytically for:
   1. ∥u∥ = 6,𝜃_u = 60^∘; ∥v∥ = 10,𝜃 _v = 120^∘; ∥w∥ = 8,𝜃 _w = 270^∘.
   2. Three unit vectors at 0^∘, 120^∘, 240^∘.
   3. u = 2î + 3ĵ, v = −5î + 2ĵ, w = −ĵ.
   4. u = î, v = −ĵ, w = î −ĵ.
2. Add
   

   and find the resultant magnitude and direction cosines.

3. Given A = (−1, 2,−1), B = (−3, 6, 6), C = (4, 3, 1), D = (0, 0, 2), find the length and direction cosines of + + .
4. Using the same points: find the point dividing AB in the ratio 2 : 1; find the centroid of A,B,C; and show that the side midpoints of the skew quadrilateral ABCD form a parallelogram.
5. Prove
   
6. Deduce
   
7. Find the Cartesian equations of the line through A = (−2, 0, 2) and B = (2, 1,−3).

Modern notation references

The notation and terminology in this modernized article follow standard present-day mechanics and vector-analysis usage, particularly:

1. J. R. Taylor, Classical Mechanics, University Science Books, 2005.
2. D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
3. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison–Wesley, 2002.

Source

This article is a modernized restatement of the corresponding Public Domain article in Louis Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.” The original 1930 edition is the source basis.