The Inner Product

Try this

1. Make the angle between $\vec{b}$ and $\vec{a}$ acute, right and obtuse, and look at the sign of the dot product each time.
2. Keep $\vec{b}$ fixed and make $\vec{a}$ twice as long (same direction). What happens to the dot product? Does the shadow change?
3. Let $\vec{b}=\vec{a}$. What is the dot product?
4. The two lines of calculation on the right use “length × shadow” and the coordinates $x_1{x}_2+y_1{y}_2$. Are they always equal?

For two nonzero vectors $\vec{a}$, $\vec{b}$ with angle $\theta$ between them ($0^{\circ }\le \theta \le 180^{\circ }$), the inner product (dot product) is

$|\vec{b}|\cos \theta$ is the projection of $\vec{b}$ in the direction of $\vec{a}$: the signed “shadow” that $\vec{b}$ casts on the line of $\vec{a}$ when the light shines perpendicular to that line. By convention the zero vector has dot product $0$ with every vector. Some useful facts:

• $\vec{a}\cdot \vec{a}=|\vec{a}{|}^2$;
• $\vec{a}\perp \vec{b}⟺\vec{a}\cdot \vec{b}=0$;
• $\cos \theta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}$, so $|\vec{a}\cdot \vec{b}|\le |\vec{a}||\vec{b}|$;
• in coordinates: if $\vec{a}=(x_1,y_1)$ and $\vec{b}=(x_2,y_2)$, then $\vec{a}\cdot \vec{b}=x_1{x}_2+y_1{y}_2$.

Why: where the coordinate formula comes from

Using the dot product to find a minimum

Expanding $|\vec{b}-t\vec{a}{|}^2$ gives a quadratic function of $t$. Here the dot product turns “shortest distance” in geometry into “completing the square”.

Practice