Equivalence of Definitions of Dot Product

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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:

\(\ds \mathbf a \cdot \mathbf b\) \(:=\) \(\ds \sum_{k \mathop = 1}^n a_k b_k\)
\(\ds \mathbf b\) \(=\) \(\ds a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i \mathop = 1}^n a_i b_i\)


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

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