The dot product

The opening pair $\mathbf{u} = (3, 1)$, $\mathbf{v} = (1, 2.5)$ gives $3(1) + 1(2.5) = 5.50$ — and the readout $|u||v|\cos\theta$ shows $5.50$ as well, at $\theta = 49.8°$. Press set orthogonal: $\mathbf{v}$ jumps to $(-0.85, 2.55)$, the readout $u \cdot v$ shows $0.00$, $\theta$ reads $90.0°$, and the proj segment shrinks to a point. Press set opposed and $u \cdot v$ swings to $-8.50$. The component recipe tracked the angle the whole time.

That is the whole surprise. The dot product has two faces — a recipe on components, and a statement about geometry — and they are always the same number:

Why they agree: dotting a vector with itself gives $v_1^2 + v_2^2 = \|\mathbf{v}\|^2$, which is Pythagoras. Expand the tip-to-tip distance $\|\mathbf{u} - \mathbf{v}\|^2$ with dot-product algebra, expand it again with the law of cosines, cancel the matching terms, and the two faces collapse into one fact. Since $\cos\theta$ carries the sign, the dot product is an alignment meter: positive for acute angles, negative for obtuse, and exactly zero at $90°$ — the vectors are orthogonal. The dotted segment in the instrument is the projection, the shadow $\mathbf{u}$ casts on the line through $\mathbf{v}$: the part of $\mathbf{u}$ that acts along $\mathbf{v}$, marked with a right-angle tick at its foot.