To find the angle between two vectors using the inner product, you utilize the fundamental relationship that connects the inner product, the norms (lengths) of the vectors, and the cosine of the angle between them.
The angle $\theta$ between two non-zero vectors $\mathbf{u}$ and $\mathbf{v}$ in an inner product space can be found using the following formula derived from the definition of the inner product:
$\cos(\theta) = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{||\mathbf{u}|| \cdot ||\mathbf{v}||}$
Where:
- $\langle \mathbf{u}, \mathbf{v} \rangle$ represents the inner product of vectors $\mathbf{u}$ and $\mathbf{v}$.
- $||\mathbf{u}||$ represents the norm (or length) of vector $\mathbf{u}$.
- $||\mathbf{v}||$ represents the norm (or length) of vector $\mathbf{v}$.
Once you calculate the value of $\cos(\theta)$, you can find the angle $\theta$ by taking the inverse cosine (arccosine):
$\theta = \arccos\left( \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{||\mathbf{u}|| \cdot ||\mathbf{v}||} \right)$
The angle $\theta$ is typically taken to be in the range $[0, \pi]$ radians (or $[0, 180^\circ]$ degrees).
Finding the angle between two vectors using the inner product involves a few key steps:
- Calculate the Inner Product: Compute $\langle \mathbf{u}, \mathbf{v} \rangle$. The specific calculation depends on the definition of the inner product for the vector space you are working in. For standard Euclidean space ($\mathbb{R}^n$), the inner product (dot product) is $\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \dots + u_nv_n$.
- Calculate the Norms: Compute $||\mathbf{u}||$ and $||\mathbf{v}||$. The norm of a vector is typically defined using the inner product as $||\mathbf{u}|| = \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle}$. In Euclidean space, this is the standard length formula: $||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2 + \dots + u_n^2}$.
- Substitute into the Formula: Plug the calculated inner product and norms into the formula for $\cos(\theta)$:
$\cos(\theta) = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{||\mathbf{u}|| \cdot ||\mathbf{v}||}$. - Find the Angle: Calculate $\theta$ by taking the arccosine of the result from Step 3:
$\theta = \arccos\left( \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{||\mathbf{u}|| \cdot ||\mathbf{v}||} \right)$.
Facts to Keep in Mind (Based on Reference)
As highlighted in discussions on inner product spaces, certain facts are crucial:
- Definition of Angle: The relationship $\langle \mathbf{u}, \mathbf{v} \rangle = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cos(\theta)$ is not derived from a geometric angle in all vector spaces; rather, it serves as the definition of the angle between two vectors in a general inner product space. This allows the concept of an angle to be extended beyond standard 2D or 3D Euclidean geometry to more abstract spaces.
- Cauchy-Schwarz Inequality: For the formula $\cos(\theta) = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{||\mathbf{u}|| \cdot ||\mathbf{v}||}$ to yield a value between -1 and 1 (so that $\arccos$ is defined), the inequality $|\langle \mathbf{u}, \mathbf{v} \rangle| \le ||\mathbf{u}|| \cdot ||\mathbf{v}||$ must hold. This is guaranteed by the Cauchy-Schwarz inequality, a fundamental theorem in inner product spaces.
- Orthogonality: If the inner product $\langle \mathbf{u}, \mathbf{v} \rangle = 0$, then $\cos(\theta) = 0$, which means $\theta = \frac{\pi}{2}$ (or $90^\circ$). This is the algebraic definition of orthogonal (perpendicular) vectors in any inner product space.
Summary Table of Symbols
| Symbol | Description |
|---|---|
| $\mathbf{u}, \mathbf{v}$ | Vectors |
| $\langle \mathbf{u}, \mathbf{v} \rangle$ | Inner product of $\mathbf{u}$ and $\mathbf{v}$ |
| $ | \mathbf{u} |
| $\theta$ | Angle between $\mathbf{u}$ and $\mathbf{v}$ |
| $\cos(\theta)$ | Cosine of the angle |
| $\arccos(x)$ | Arccosine (inverse cosine) of $x$ |
By using the inner product to define and calculate the components of the cosine formula, you can determine the angle between vectors even in spaces where direct geometric visualization is not possible.
