Dot Product Calculator

How to Calculate the Dot Product of Two Vectors

The dot product, also known as the scalar product, is a fundamental operation in vector algebra with applications in physics, engineering, and computer science. It results in a scalar value that represents how much two vectors align with each other. Here's a comprehensive guide on how to calculate the dot product of two vectors:

Dot Product Formula

2D Vectors

For two 2D vectors \(\vec{a} = (a_x, a_y)\) and \(\vec{b} = (b_x, b_y)\), their dot product \(\vec{a} \cdot \vec{b}\) is defined as:

\[ \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y \]

3D Vectors

For two 3D vectors \(\vec{a} = (a_x, a_y, a_z)\) and \(\vec{b} = (b_x, b_y, b_z)\), their dot product \(\vec{a} \cdot \vec{b}\) is defined as:

\[ \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z \]

Where:

• \(\vec{a} \cdot \vec{b}\) is the resulting dot product (a scalar)
• \(a_x, a_y, a_z\) are the components of vector \(\vec{a}\)
• \(b_x, b_y, b_z\) are the components of vector \(\vec{b}\)

Calculation Steps

1. Identify the components of both vectors
2. Multiply corresponding components
3. Sum the products of the components

Example Calculation

2D Example

Let's calculate the dot product of two 2D vectors:

\(\vec{a} = (2, 3)\) and \(\vec{b} = (5, 6)\)

1. Multiply corresponding components:
   • \(a_x b_x = 2 \times 5 = 10\)
   • \(a_y b_y = 3 \times 6 = 18\)
2. Sum the products: \[ \vec{a} \cdot \vec{b} = 10 + 18 = 28 \]

Therefore, the dot product of \(\vec{a}\) and \(\vec{b}\) is 28.

3D Example

Let's calculate the dot product of two 3D vectors:

\(\vec{a} = (2, 3, 4)\) and \(\vec{b} = (5, 6, 7)\)

1. Multiply corresponding components:
   • \(a_x b_x = 2 \times 5 = 10\)
   • \(a_y b_y = 3 \times 6 = 18\)
   • \(a_z b_z = 4 \times 7 = 28\)
2. Sum the products: \[ \vec{a} \cdot \vec{b} = 10 + 18 + 28 = 56 \]

Therefore, the dot product of \(\vec{a}\) and \(\vec{b}\) is 56.

Visual Representation

This diagram illustrates two 2D vectors \(\vec{a}\) and \(\vec{b}\). The dot product is related to the angle θ between these vectors by the formula:

\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]

Where \(|\vec{a}|\) and \(|\vec{b}|\) are the magnitudes of the vectors, and θ is the angle between them.