What is a dot product?

Key takeaways:

• The dot product calculates the product of two vectors and is used in physics, engineering, and computer science.

• The dot product can be defined in two different ways:

  • Algebraic representation of dot product: x·y = (x1 × y1) + (x2 × y2)

  • Geometric representation of dot product: x·y = |x||y|cos(θ)

• Properties of dot product:

  • Commutative property: The order of multiplication doesn't affect the result: x⋅y = y⋅x

  • Distributive property: Multiplication distributes over addition x⋅(y+z) = x⋅y+x⋅z

  • Scalar multiplication: Scaling a dot product can be done on either vector: c(x⋅y) = (cx)⋅y = x⋅(cy)

  • Orthogonal vectors: If two vectors are perpendicular, their dot product is zero: x⋅y = 0

• Applications of dot product: It is used in fields like physics for calculating work done, in geometry to find angles between vectors, and in computer graphics for lighting and projection calculations.

Dot product

The dot product, also known as the scalarA scalar is a quantity in mathematics that has magnitude only and no direction, unlike vectors, which have both magnitude and direction. product, computes the product of two vectors and gives a scalar value that represents how much two vectors align in the same direction. It quantifies the magnitude of one vector projected onto the other. If the vectors point in the same direction, the dot product is positive and large; if they are perpendicular, the dot product is zero; and if they point in opposite directions, the dot product is negative. It is widely used in fields like physics, engineering, and computer science, particularly in understanding projections, magnitudes, and calculating work done in force fields.

What does dot product tell you?

The dot product measures how much two vectors align. Suppose you're walking along a street, and a breeze is blowing. You want to know how much of that breeze is helping push you forward versus how much is just blowing past. The direction you're walking is one vector, and the direction of the breeze is another.

If the breeze blows directly behind you, it’s fully pushing you forward, and the dot product is at its maximum. If it blows sideways at a 90-degree angle, there's no forward push, and the dot product is zero. The dot product tells you how much these directions align, with higher values indicating greater alignment.

Now, we'll take an example of two vectors $x$ and $y$, the below diagram shows these vectors and angle between them:

The dot product can be defined in two different ways: algebraically and geometrically. Let's explore both approaches.

Algebraic definition of dot product

Algebraically, the dot product of two vectors is the summation of the products of their corresponding components. For two vectors $x$ and $y$, the dot product is defined as:

Example

Let's consider two vectors, $x = [-6, 8]$ and $y = [5, 12].$

The dot product of these vectors is calculated as:

Algebraical implementation of dot product

Here’s a simple code implementation of the dot product between two vectors using NumPy in Python:

This calculation illustrates how easily we can compute the dot product of vectors in Python. The result is a scalar, highlighting the reason why the dot product is also called the scalar product.

Geometric definition of dot product

Geometrically, the dot product represents the product of the magnitudes of two vectors and the cosine of the angle between them. For two vectors $x$ and $y$, the dot product can be written as:

Where:

• $|x|$ and $|y|$  are the magnitudes of vectors $x$ and $y$, respectively.

• $θ$ is the angle between the vectors.

Magnitude of a vector

The magnitude (or length) of a vector $x$ is computed as:

Angle between two vectors

The angle $θ$ between vectors $x$ and $y$ can be determined using the following formula:

Example

Let's consider two vectors $x = [-6, 8]$ and $y = [5, 12].$To compute their dot product, we first calculate the magnitudes:

As the angle $θ$ between them is $59.5°$ then,

Thus the dot product of $x$ and $y$ is:

Geometrical implementation of dot product

Below is an implementation of the dot products between two vectors, taking into account their magnitudes.

Properties of dot product

The dot product holds several important properties that make it a fundamental operation in vector algebra. Let $x$, $y$, and $z$ be vectors and $c$ be a scalar. The properties governing their dot products are listed in the table below.

Applications of dot product

The dot product is used in many fields, including:

• Physics: Calculating work done by a force along a displacement uses the dot product of the force and displacement vectors.

• Geometry: Understanding the angle between vectors and determining if vectors are perpendicular (orthogonal).

• Computer graphics: In 3D rendering, the dot product is crucial for lighting calculations and projections.

Conclusion

In conclusion, the dot product is a versatile vector operation used across various fields. It helps understand vector relationships, such as magnitudes and angles, and is essential for solving mathematical problems and practical applications like computing work in force fields and 3D graphics rendering.