How large can an inner product be?

Source scope: §3.5.1–3.5.2. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

When do squared lengths add?

For $(3,0)$ and $(0,4)$, the sum has squared length $25=9+16$. For $(3,0)$ and $(4,0)$, it has squared length $49$. The missing condition is orthogonality.

Theorem 1 (source). In a real inner product space, if $x\perp y$, then

Source proof. Expand using linearity and symmetry:

Both cross terms vanish because the vectors are orthogonal. This proof works in any real inner product space, not just a drawn triangle.

Alignment has a limit

The Cauchy–Schwarz inequality says

Equality holds exactly when the pair is linearly dependent. This includes the case where either vector is zero. For two nonzero vectors it means that one is a scalar multiple of the other. This wording also covers a zero vector paired with a nonzero vector without requiring the nonzero vector to be a multiple of zero.

The source states the inequality and equality condition without proof. Here is a teaching proof. If $y=0$, both sides are zero. Otherwise set $a=\langle x,y\rangle/\|y\|^2$. Positivity gives

Multiply by the positive denominator and take square roots. Equality holds precisely when $x-ay=0$. For $(1,2)$ and $(2,4)$, both sides equal $10$. For $(1,0)$ and $(0,1)$, the left side is zero and the right side is one.

Transfer

Can the absolute dot product of vectors with lengths two and three equal seven?