Inner Product¶

Definition¶

For vectors $\boldsymbol{x}=(x_1,\ldots,x_n)$ and $\boldsymbol{y}=(y_1,\ldots,y_n)$ in $\mathbb{R}^n$, the dot product, or standard inner product, is

More generally, an inner product on a real vector space $V$ is a function $\langle\cdot,\cdot\rangle\colon V\times V\to\mathbb{R}$ that is

• symmetric: $\langle \boldsymbol{x},\boldsymbol{y}\rangle=\langle \boldsymbol{y},\boldsymbol{x}\rangle$$⟨\mathbf{y},\mathbf{x}⟩$;
• linear in the first argument, $\langle \alpha \boldsymbol{x}+\beta \boldsymbol{y},\boldsymbol{z}\rangle=\alpha\langle \boldsymbol{x},\boldsymbol{z}\rangle+\beta\langle \boldsymbol{y},\boldsymbol{z}\rangle$$\beta ⟨\mathbf{y},\mathbf{z}⟩$, and hence, by symmetry, in the second;
• positive definite: $\langle \boldsymbol{x},\boldsymbol{x}\rangle>0$ for every $\boldsymbol{x}\neq\boldsymbol{0}$$0$.

The dot product is an inner product on $\mathbb{R}^n$, and it is the one meant when Math II speaks of the inner product of two vectors in $\mathbb{R}^n$.

Interpretation¶

The inner product combines two vectors into one number that measures how much they point in the same direction. Length, distance, angle and orthogonality are all defined from it.

Dropping a perpendicular from the tip of $\boldsymbol{y}$ onto the line of $\boldsymbol{x}$ gives the green segment of signed length $\|\boldsymbol{y}\|\cos\theta$, where $\theta$ is the angle between the vectors and $\|\cdot\|$ the Euclidean norm. The dot product is $\|\boldsymbol{x}\|$ times this signed length, $\boldsymbol{x}\cdot \boldsymbol{y}=\|\boldsymbol{x}\|\,\|\boldsymbol{y}\|\cos\theta$$\Vert \mathbf{x}\Vert \Vert \mathbf{y}\Vert \cos \theta$: positive for an acute angle as drawn, zero for a right angle, and negative for an obtuse angle, when the foot falls on the other side of the origin.

Each property in the general definition has a role: symmetry says the order does not matter; linearity, together with symmetry, lets inner products be expanded like products of numbers, as in $\|\boldsymbol{x}+\boldsymbol{y}\|^2=\|\boldsymbol{x}\|^2+2\,\boldsymbol{x}\cdot \boldsymbol{y}+\|\boldsymbol{y}\|^2$$\Vert \mathbf{y}{\Vert }^2$; positive definiteness gives every nonzero vector a positive length.

The Cauchy-Schwarz inequality and the triangle inequality follow from these properties alone, so they hold for every inner product. Dot products are also hidden in every matrix product: entry $(i,j)$ of $\boldsymbol{A}\boldsymbol{B}$ is the dot product of row $i$ of $\boldsymbol{A}$ with column $j$ of $\boldsymbol{B}$.

A common slip is to multiply entry by entry and stop there: the inner product is a number, not a vector, so $(1,2)\cdot(3,4)=11$$11$, not $(3,8)$.

Key Results¶

• $\boldsymbol{x}\cdot \boldsymbol{x}\geq0$$0$, and $\boldsymbol{x}\cdot \boldsymbol{x}=0$ if and only if $\boldsymbol{x}=\boldsymbol{0}$$0$.
• For real vectors, $\boldsymbol{x}\cdot \boldsymbol{y}=\boldsymbol{y}\cdot \boldsymbol{x}$$\mathbf{x}$.
• Together with linearity, these are the defining properties of an inner product, so everything proved from them alone, such as the Cauchy-Schwarz inequality, holds for every inner product.

Examples¶

• $(1,2)\cdot(3,4)=11$$11$.
• $(1,0)\cdot(0,1)=0$$0$, so the vectors are orthogonal.
• $\langle \boldsymbol{x},\boldsymbol{y}\rangle=2x_1y_1+x_2y_2$ is an inner product on $\mathbb{R}^2$ different from the dot product.

Prerequisites¶

• Vector

Related Concepts¶

• Euclidean Norm
• Idempotent Matrix
• Variance and Standard Deviation
• Coordinate
• Euclidean Space

Sources¶

• Math II: Linear Algebra · Week 5
• Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: Euclidean inner product Keywords: dot product, standard inner product, scalar product, inner product space Symbols: alpha, beta, real numbers, theta

Last updated: 27 October 2026 11:52