Dot Products, Length and Angle

Definition For vectors $\mathbf{x} = (x_1, \dots, x_n)$ and $\mathbf{y} = (y_1, \dots, y_n)$ in $\mathbb{R}^n$, the dot product is the real number

The row $\mathbf{x}^{\top}$ times the column $\mathbf{y}$ is a $1 \times 1$ matrix, identified with that number. The dot product is defined only when both vectors have the same number of entries. For all $\mathbf{x}, \mathbf{y}, \mathbf{z}$ in $\mathbb{R}^n$ and every real $c$ it satisfies

and $\mathbf{x} \cdot \mathbf{x} = x_1^2 + \cdots + x_n^2 \ge 0$, with equality only for $\mathbf{x} = \mathbf{0}$.

Definition The Euclidean length, or norm, of $\mathbf{x} = (x_1, \dots, x_n)$ in $\mathbb{R}^n$ is

It satisfies $\lVert \mathbf{x} \rVert \ge 0$, with $\lVert \mathbf{x} \rVert = 0$ only for $\mathbf{x} = \mathbf{0}$, and $\lVert c\mathbf{x} \rVert = \lvert c \rvert \, \lVert \mathbf{x} \rVert$ for every real $c$. In $\mathbb{R}^2$ and $\mathbb{R}^3$ it is the length given by the Pythagorean theorem, applied once in the plane and once more for the third coordinate.

Definition A unit vector is a vector of length $1$. For a nonzero vector $\mathbf{x}$, the vector

is the unit vector in the direction of $\mathbf{x}$. Producing it is called normalizing $\mathbf{x}$. The zero vector cannot be normalized, since its length is $0$ and it has no direction.

Definition The distance between vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^n$ is $\lVert \mathbf{x} - \mathbf{y} \rVert$, the length of their difference. It equals $\lVert \mathbf{y} - \mathbf{x} \rVert$. Expanding the squares entry by entry gives

because each entry contributes $(x_i - y_i)^2 = x_i^2 - 2x_i y_i + y_i^2$.

Definition Vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^n$ are orthogonal if $\mathbf{x} \cdot \mathbf{y} = 0$. By the expansion of the squared distance, this happens exactly when

which for nonzero vectors in $\mathbb{R}^2$ or $\mathbb{R}^3$ is the Pythagorean condition for a right angle between them. So for nonzero vectors in the plane or in space, orthogonal means perpendicular. In higher dimensions the dot product condition is the definition. The zero vector is orthogonal to every vector.

Definition For nonzero vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^n$, the angle between them is the unique $\theta$ with $0 \le \theta \le \pi$ such that

In $\mathbb{R}^2$ and $\mathbb{R}^3$ this agrees with the geometric angle. The law of cosines for the triangle with corners $\mathbf{0}$, $\mathbf{x}$ and $\mathbf{y}$ gives

and comparing with the expansion of the squared distance leaves $\mathbf{x} \cdot \mathbf{y} = \lVert \mathbf{x} \rVert \, \lVert \mathbf{y} \rVert \cos \theta$. In higher dimensions the formula is the definition. It makes sense because the quotient always lies between $-1$ and $1$, a fact called the Cauchy-Schwarz inequality and proved in the next lesson. The angle is not defined when either vector is zero.