Vectors & Dot Products

Lengths come from v · v, and orthogonality is exactly v · w = 0.

The idea

A vector in $\mathbb{R}^n$ is an ordered list of $n$ numbers, $v = (v_1, \ldots, v_n)$. In the plane we picture $v = (a, b)$ as an arrow drawn from the origin, reaching $a$ units across and $b$ units up, as in the figure.

The coordinates determine the arrow's length. The arrow is the hypotenuse of a right triangle with legs $a$ and $b$, so the Pythagorean theorem gives the length $\sqrt{a^{2} + b^{2}}$, and the same formula in $n$ dimensions defines the norm. The dot product of two vectors multiplies their matching coordinates and adds the results.

Definition (Norm and dot product).

The norm of $v = (v_1, \ldots, v_n)$ is $\|v\| = \sqrt{v_1^{2} + \cdots + v_n^{2}},$ and the dot product of $v$ with $w = (w_1, \ldots, w_n)$ is $v \cdot w = v_1 w_1 + \cdots + v_n w_n.$

The dot product is a single number, not a vector, and it measures relative direction: it is positive when the vectors point broadly the same way, negative when they point against each other, and zero exactly when they are perpendicular. Angle, perpendicularity and projection all come out of this one formula.

Ways to work on it

• Walkthrough. Compute lengths and dot products, and test when two vectors are perpendicular.
• Practice. Compute random 2D dot products and read off the geometry from the sign.
• Hardest. Construct a vector orthogonal to a given one.

Not sure where to start? Take the ten-question placement test.