Norms and Orthogonality: Inner Products Made Simple

A norm assigns every vector a nonnegative length ||v|| with three rules: only zero has length zero, stretching by c multiplies length by |c|, and the triangle inequality ||u+v|| is at most ||u||+||v||. Euclidean length is one norm, but the axioms admit many others, each measuring size differently.

Orthogonality is its companion: vectors with inner product zero, the abstract right angle. Together they give Pythagoras in any inner product space, decompose vectors into independent components, and explain why perpendicular directions never interfere, the principle behind GPS triangulation and noise-cancelling headphones.

Key Concepts

• Norm: ||v|| at least 0 with equality only at 0; ||cv|| = |c| ||v||; ||u+v|| at most ||u||+||v||.
• Orthogonal / orthonormal: (u,v) = 0; plus unit norm for orthonormal.
• Normalisation: v/||v|| has norm 1 for nonzero v.
• Common norms on R to the n: Euclidean 2-norm; 1-norm, sum of absolute values; infinity-norm, max absolute value.

Formulas

• Pythagoras: (u,v) = 0 implies ||u+v|| squared = ||u|| squared + ||v|| squared.
• Normalisation: v/||v|| is the unit vector in v's direction.
• Parallelogram law: ||u+v|| squared + ||u minus v|| squared = 2||u|| squared + 2||v|| squared, true exactly for inner-product norms.
• Angle: cos(theta) = (u,v)/(||u|| ||v||).

Worked Example

Take u = (1,2), v = (2,minus 1) with the dot product: (u,v) = 2 minus 2 = 0, so orthogonal. Norms: ||u|| = ||v|| = square root of 5. Pythagoras check: u+v = (3,1), ||u+v|| squared = 10, and ||u|| squared + ||v|| squared = 5+5 = 10, matching. Normalising: u/||u|| = (1/square root of 5, 2/square root of 5) has squared norm 1/5+4/5 = 1. Contrast norms: the 1-norm of u is 3 and infinity-norm is 2, versus Euclidean 2.236, same vector, different measures.

Common Misconceptions

• Orthogonal vectors look perpendicular. Orthogonality is (u,v) = 0, which under weighted products can look slanted.
• All norms come from inner products. The 1-norm fails the parallelogram law: with (1,0),(0,1) it gives 8 versus 4.
• Pythagoras needs a triangle drawing. Expanding (u+v,u+v) kills the cross terms algebraically; no picture needed.

Real-World Applications

• GPS: fixes combine ranges along nearly orthogonal satellite directions so errors do not amplify.
• Audio: noise cancellation projects sound onto the subspace orthogonal to the noise.
• Machine learning: 1-norm regularisation encourages sparse, interpretable models.

Norms quantify size under three rules; orthogonality unlocks Pythagoras and clean decomposition. Test with the inner product, verify numerically, and remember only inner-product norms satisfy the parallelogram law.