Dot Product of Vectors and Properties – Example

The dot product is a fundamental operation in vector algebra that provides valuable insights into the relationship between vectors.

It enables us to calculate angles, determine orthogonality, and perform various other calculations.

Understanding the dot product is crucial in many mathematical and scientific fields.

It also known as the scalar product or inner product, is an operation defined between two vectors that results in a scalar value. It is denoted using the dot symbol ($\cdot$) or sometimes using the multiplication symbol ($\times$).

For two vectors $\mathbf{a}$ and $\mathbf{b}$ in $n$-dimensional space, the dot product is defined as:

\[
\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n = \sum_{i=1}^{n} a_ib_i
\]

where $a_i$ and $b_i$ represent the components of the vectors $\mathbf{a}$ and $\mathbf{b}$, respectively.

 

Properties

The dot product exhibits several important properties:

1. Commutativity: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
2. Distributivity over vector addition: $(\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}$
3. Scalar multiplication: $(k \mathbf{a}) \cdot \mathbf{b} = k(\mathbf{a} \cdot \mathbf{b}) = \mathbf{a} \cdot (k \mathbf{b})$
4. Orthogonality: If $\mathbf{a} \cdot \mathbf{b} = 0$, then $\mathbf{a}$ and $\mathbf{b}$ are orthogonal.

 

Examples

Let’s consider a couple of examples to illustrate the concept of the dot product:

• Example 1: Suppose we have two vectors $\mathbf{a} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$. To find their dot product, we multiply the corresponding components and sum the results:

\[
\mathbf{a} \cdot \mathbf{b} = (1 \cdot 4) + (2 \cdot 5) + (3 \cdot 6) = 4 + 10 + 18 = 32
\]

Therefore, $\mathbf{a} \cdot \mathbf{b} = 32$.

• Example 2: Consider two vectors $\mathbf{u} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} 4 \\ 1 \end{bmatrix}$. To determine whether these vectors are orthogonal, we calculate their dot product:

\[
\mathbf{u} \cdot \mathbf{v} = (2 \cdot 4) + (-3 \cdot 1) = 8 – 3 = 5
\]

Since $\mathbf{u} \cdot \mathbf{v} \neq 0$, we can conclude that $\mathbf{u}$ and $\mathbf{v}$ are not orthogonal.