Inner Products, Length and Orthogonality
An inner product gives a general space a notion of length and angle. Orthogonal sets are independent, and orthonormal ones make coordinates cheap to compute.
What a learner can do afterwards
- Verify the inner product axioms for a given rule
- Show that an orthogonal set of nonzero vectors is independent
- Compute coordinates in an orthonormal basis with inner products alone
- Compute dot products and lengths in R2 as the model inner product
- Decide orthogonality of small vectors from a zero inner product
1 · Read
An inner product gives a space its geometry. The dot product (a, b) dot (c, d) = ac + bd is the model: (4, 1) dot (2, 3) = 8 + 3 = 11. Length flows from it as sqrt(<v, v>), so (6, 8) has length 10. Angles use cos theta = (u dot v) / (|u| |v|), and zero means orthogonal: (3, minus 1) is orthogonal to (1, 3).
Not every rule qualifies, so test candidates against the axioms. The rule <u, v> = u1 v1 minus u2 v2 fails positivity: <(0, 2), (0, 2)> = minus 4. Breaking one axiom on purpose teaches the definition faster than reading it. Never trust a rule you have not verified.
Orthogonal sets of nonzero vectors are always independent. Suppose a dependence relation holds; dotting it with each member forces every coefficient to zero, since each <vj, vj> > 0. That one argument is the whole second skill in this topic.
Orthonormal bases make coordinates cheap: dot v with each basis vector, no system solving. With e1 = (3/5, 4/5), e2 = (minus 4/5, 3/5) and v = (2, 1), the first coordinate is 6/5 + 4/5 = 10/5 = 2 and the second is minus 8/5 + 3/5 = minus 1. Rewrite any length computation as sqrt(<v, v>) and the pattern sticks.
Test the axioms, turn dots into length and right angles, and read orthonormal coordinates straight off the inner product.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.