Direction Cosines

Trigonometry · Axiom Academy

Understanding the angles a vector makes with coordinate axes and the fundamental identity cos²α + cos²β + cos²γ = 1 1. What Are Direction Cosines? Consider a vector v in 3D space. This vector makes three angles: The direction cosines have a beautiful geometric interpretation: they represent the projections of a unit vector onto each coordinate axis. If our vector has magnitude r and components (x, y, z), then: Here's the remarkable property: for any vector in 3D space, the sum of the squares of its direction cosines always equals 1. This isn't just a coincidence—it's a direct consequence of the Pythagorean theorem in three dimensions! Let's see why cos²α + cos²β + cos²γ = 1 must be true. Consider a vector with components (x, y, z) and magnitude r: The direction cosines are: cos α = x/r, cos β = y/r, cos γ = z/r Squaring each: cos²α = x²/r², cos²β = y²/r², cos²γ = z²/r² Adding them: cos²α + cos²β + cos²γ = (x² + y² + z²)/r² But by the 3D Pythagorean theorem: x² + y² + z² = r² Therefore: cos²α + cos²β + cos²γ = r²/r² = 1 ✓

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