How does a product measure alignment?

Source scope: §3.5. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Add the coordinate contributions

For $x=(2,1)$ and $y=(1,-2)$, the products are $2$ and $-2$. Their sum is zero. The vectors are perpendicular even though neither coordinate product is zero.

An inner product on a real vector space is a function $\langle\cdot,\cdot\rangle:V\times V\to\mathbb R$ satisfying

1. $\langle x,x\rangle\ge0$, with equality exactly when $x=0$.
2. Linearity in the first slot: $\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle$ and $\langle\alpha x,y\rangle=\alpha\langle x,y\rangle$.
3. Symmetry: $\langle x,y\rangle=\langle y,x\rangle$.

These hold for all vectors and real scalars. Symmetry also supplies linearity in the second slot. A space equipped with an inner product is an inner product space.

From alignment to length

The induced norm is $\|x\|=\sqrt{\langle x,x\rangle}$. Thus an inner product gives a norm and, in turn, a metric. Positivity and absolute homogeneity follow from the axioms. The triangle inequality uses Cauchy–Schwarz from the next lesson:

Vectors are orthogonal, written $x\perp y$, when $\langle x,y\rangle=0$. Orthogonal unit vectors are orthonormal. Zero is orthogonal to every vector but is never a unit vector.

The standard real inner product is

The matrix product is $1\times1$, which we identify with a scalar. Its induced norm is exactly the two-norm because $x^Tx=\sum_i x_i^2$. For nonzero Euclidean vectors, $x^Ty=\|x\|_2\|y\|_2\cos\theta$ connects the product to the angle.

Source footnote 5: A space complete in the metric induced by its inner product is a Hilbert space.

Transfer

If two vectors have dot product zero, must both have length one?