Inner product on ℝ^k

The dot product ⟨x,y⟩ = sum xi yi defining angles and lengths in ℝ^k.

Inner product on ℝ^k

For $x=(x_1,\dots,x_k)$ and $y=(y_1,\dots,y_k)$ in $\mathbb{R}^k$ , the (standard) inner product is

\langle x,y\rangle := \sum_{i=1}^k x_i y_i.

The inner product is bilinear, symmetric, and positive definite, and it generates the Euclidean norm by $\|x\|_2=\sqrt{\langle x,x\rangle}$ . It is the algebraic structure behind orthogonality, projections, and many inequalities.

Examples:

In $\mathbb{R}^2$ , $\langle (1,2),(3,4)\rangle = 1\cdot 3 + 2\cdot 4 = 11$ .
$\langle x,x\rangle \ge 0$ for all $x$ , with equality iff $x=0$ .
If $x=(1,0)$ and $y=(0,1)$ , then $\langle x,y\rangle=0$ .