## Quick Math Dot Product Matrix

##### Enter expression in this form

Example (3x3 Matrix): Enter expression as [3,4,5],[6,7,8]

### Dot Product, Matrix

In vector algebra, the dot product (also known as the scalar product) is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This number represents the product of the vectors’ magnitudes and the cosine of the angle between them. The dot product is a fundamental operation in various fields, including physics, engineering, and computer science.

#### Definition

Given two vectors \(\mathbf{a} = [a_1, a_2, a_3]^T\) and \(\mathbf{b} = [b_1, b_2, b_3]^T\), their dot product is defined as:

\[

\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3

\]

#### Properties of the Dot Product

1. Commutative: \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\)

2. Distributive: \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}\)

3. Scalar Multiplication: \((c\mathbf{a}) \cdot \mathbf{b} = c(\mathbf{a} \cdot \mathbf{b})\) for any scalar \(c\)

#### Example

Let’s calculate the dot product of \(\mathbf{a} = [1, 2, 3]^T\) and \(\mathbf{b} = [4, 5, 6]^T\).

**1. Apply the Dot Product Formula**:

\[

\mathbf{a} \cdot \mathbf{b} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6

\]

**2. Perform the Multiplications and Addition:**

\[

\mathbf{a} \cdot \mathbf{b} = (1 \cdot 4) + (2 \cdot 5) + (3 \cdot 6)

= 4 + 10 + 18

= 32

\]

#### Interpretation

The dot product \( \mathbf{a} \cdot \mathbf{b} = 32 \) represents the sum of the products of the corresponding entries of the vectors \(\mathbf{a}\) and \(\mathbf{b}\). It can also be interpreted geometrically as:

\[

\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta

\]

where \(\|\mathbf{a}\|\) and \(\|\mathbf{b}\|\) are the magnitudes (lengths) of vectors \(\mathbf{a}\) and \(\mathbf{b}\), respectively, and \(\theta\) is the angle between them.

#### Verification with Magnitudes and Angle

To verify using magnitudes and the cosine of the angle between the vectors:

**1. Calculate Magnitudes:**

\[

\|\mathbf{a}\| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}

\]

\[

\|\mathbf{b}\| = \sqrt{4^2 + 5^2 + 6^2} = \sqrt{16 + 25 + 36} = \sqrt{77}

\]

**2. Dot Product in Terms of Magnitudes and Angle:**

\[

\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta

\]

Given \(\mathbf{a} \cdot \mathbf{b} = 32\), we have:

\[

32 = \sqrt{14} \cdot \sqrt{77} \cdot \cos \theta

\]

Solving for \(\cos \theta\), we get:

\[

\cos \theta = \frac{32}{\sqrt{14 \cdot 77}} = \frac{32}{\sqrt{1078}} \approx 0.975

\]

This verifies our dot product calculation and provides insight into the geometric relationship between the vectors.

The dot product is a powerful tool for measuring the similarity and angular relationship between vectors. By understanding both its algebraic and geometric interpretations, we gain deeper insights into vector interactions in various applications.