Angle Between Vectors¶

Theorem¶

For nonzero vectors $\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^n$${\mathbb{R}}^n$, the angle $\theta$ between them satisfies $\boldsymbol{x}\cdot \boldsymbol{y}=\|\boldsymbol{x}\|\|\boldsymbol{y}\|\cos\theta$$\Vert \mathbf{x}\Vert \Vert \mathbf{y}\Vert \cos \theta$.

Interpretation¶

The angle formula turns the algebraic inner product into geometry. Dividing $\boldsymbol{x}\cdot \boldsymbol{y}$ by the Euclidean norms $\|\boldsymbol{x}\|$ and $\|\boldsymbol{y}\|$ removes the lengths and leaves only the information about direction, $\cos\theta$.

For the vectors drawn, $\cos\theta=\dfrac{\boldsymbol{x}\cdot \boldsymbol{y}}{\|\boldsymbol{x}\|\,\|\boldsymbol{y}\|}=\dfrac{4+3}{\sqrt{17}\sqrt{10}}\approx0.54$$0.54$, so $\theta\approx57.5^\circ$$57.5^{\circ }$. Two nonzero vectors are perpendicular when their inner product is zero and parallel when their angle is $0$ or $180^\circ$.

In $\mathbb{R}^2$ and $\mathbb{R}^3$ the formula follows from the law of cosines in the triangle with sides $\boldsymbol{x}$, $\boldsymbol{y}$ and $\boldsymbol{x}-\boldsymbol{y}$$\mathbf{y}$. In $\mathbb{R}^n$ with $n>3$ there is no picture to measure in, and the formula is used to define $\theta\in[0,\pi]$ as the inverse cosine of the quotient. This definition makes sense because the Cauchy-Schwarz inequality guarantees that the quotient lies between $-1$ and $1$.

Because of the normalization, changing the length of one vector does not change the angle: replacing $\boldsymbol{x}$ by $2\boldsymbol{x}$ doubles both $\boldsymbol{x}\cdot \boldsymbol{y}$ and $\|\boldsymbol{x}\|$. A negative multiple is different: replacing $\boldsymbol{x}$ by $-\boldsymbol{x}$ changes the sign of $\cos\theta$ and turns $\theta$ into $\pi-\theta$$\theta$. Since the cosine is positive on $[0,\pi/2)$ and negative on $(\pi/2,\pi]$, the sign of $\boldsymbol{x}\cdot \boldsymbol{y}$ alone tells acute from obtuse.

The formula needs nonzero vectors. For $\boldsymbol{x}=\boldsymbol{0}$ the quotient is undefined and no angle is assigned, although $\boldsymbol{0}\cdot \boldsymbol{y}=0$ makes the zero vector orthogonal to every vector.

Key Results¶

• If $\boldsymbol{x}\cdot \boldsymbol{y}>0$$0$, the angle is acute.
• If $\boldsymbol{x}\cdot \boldsymbol{y}=0$$0$, the angle is right and the vectors are orthogonal.

Examples¶

• $(1,0)$ and $(0,1)$ have angle $\pi/2$.
• $(1,0)$ and $(1,1)$ have a positive inner product, so their angle is acute.

Prerequisites¶

• Inner Product
• Euclidean Norm

Related Concepts¶

• Orthogonal Vectors
• Cauchy-Schwarz Inequality
• Euclidean Space

Sources¶

• Math II: Linear Algebra · Week 5

Aliases: angle between two vectors Keywords: vector angle Symbols: element of, pi, real numbers, square root, theta

Last updated: 20 October 2026 03:02