Equivalence of Definitions of Dot Product
Theorem
The following definitions of the concept of Dot Product in the context of Real Euclidean Space are equivalent:
Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^n$.
General Context
The dot product of $\mathbf a$ and $\mathbf b$ is defined as:
\(\ds \mathbf a \cdot \mathbf b\) | \(:=\) | \(\ds \sum_{k \mathop = 1}^n a_k b_k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i \mathop = 1}^n a_i b_i\) |
If the vectors are represented as column matrices:
- $\mathbf a = \begin {bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end {bmatrix} , \mathbf b = \begin {bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end {bmatrix}$
we can express the dot product as:
- $\mathbf a \cdot \mathbf b = \mathbf a^\intercal \mathbf b$
where:
- $\mathbf a^\intercal = \begin {bmatrix} a_1 & a_2 & \cdots & a_n \end {bmatrix}$ is the transpose of $\mathbf a$
- the operation between the matrices is the matrix product.
Definition by Cosine
Let $\mathbf a$ and $\mathbf b$ be vectors in real Euclidean space $\R^n$.
The dot product of $\mathbf a$ and $\mathbf b$ is defined as:
- $\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \angle \mathbf a, \mathbf b$
where:
- $\norm {\mathbf a}$ denotes the length of $\mathbf a$
- $\angle \mathbf a, \mathbf b$ is the angle between $\mathbf a$ and $\mathbf b$, taken to be between $0$ and $\pi$.
Proof
General Context implies Definition by Cosine
Let $\mathbf a \cdot \mathbf b$ be a dot product in its general context.
From Cosine Formula for Dot Product:
- $\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the angle between $\mathbf v$ and $\mathbf w$.
Thus $\cdot$ is a dot product by cosine definition.
$\Box$
Definition by Cosine implies General Context
Let $\mathbf a \cdot \mathbf b$ be a dot product by cosine definition.
Let $\mathbf a$ and $\mathbf b$ be expressed in the form:
\(\ds \mathbf a\) | \(=\) | \(\ds a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_n \mathbf e_n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n a_i \mathbf e_i\) | ||||||||||||
\(\ds \mathbf b\) | \(=\) | \(\ds b_1 \mathbf e_1 + b_2 \mathbf e_2 + \cdots + b_n \mathbf e_n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^n b_j \mathbf e_j\) |
Then we have:
\(\ds \mathbf a \cdot \mathbf b\) | \(=\) | \(\ds \paren {\sum_{i \mathop = 1}^n a_i \mathbf e_i} \cdot \paren {\sum_{j \mathop = 1}^n b_j \mathbf e_j}\) | |||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i, j \mathop = 1}^n a_i b_j \mathbf e_i \mathbf e_j\) | Dot Product Distributes over Addition | ||||||||||||
noting at this point that Dot Product Distributes over Addition has been derived from Definition of Dot Product | |||||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i, j \mathop = 1}^n a_i b_j \delta_{i j}\) | Dot Product of Orthonormal Basis Vectors | ||||||||||||
noting at this point that Dot Product of Orthonormal Basis Vectors has also ultimately been derived from Definition of Dot Product | |||||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n a_i b_i\) | simplifying |
Thus $\cdot$ is a dot product in its general context.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $2$. The Scalar Product: $(2.10)$
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 2$.
- 1992: Frederick W. Byron, Jr. and Robert W. Fuller: Mathematics of Classical and Quantum Physics ... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product: $(1.3)$