Welcome back. In our last session, you mastered the fundamental operations of vector addition and scalar multiplication—operations that take vectors and produce new vectors. We saw how these can be understood both as geometric transformations and as straightforward numerical calculations.
Today, we introduce a new kind of "multiplication" between two vectors. This operation, called the dot product, is different in a crucial way: instead of producing another vector, it results in a single number, a scalar. This simple calculation is the bridge that connects the algebra of vectors to the geometry of length and angles. By the end of this lesson, you will be able to calculate the dot product of any two vectors.
A New Kind of Product
So far, when we've combined vectors, like in , the result has always been another vector. The dot product is a different concept. It takes two vectors, "multiplies" them in a specific way, and the output is a scalar. This is a powerful idea that will allow us to measure geometric properties.
Let's begin with a clear introduction to what the dot product is and how it's calculated from Khan Academy.
Vector dot product and vector length | Vectors and spaces | Linear Algebra | Khan Academy
This video, "Vector dot product and vector length," introduces the definition of the dot product.
Please watch from the introduction of the topic at this segment. Focus on three key points: The notation for the dot product (\vec{a} \cdot \vec{b}). The formula for calculating it: multiplying corresponding components and then summing them up. The fact that the result is a scalar, not a vector.
The Definition and Calculation
As the video explained, the dot product of two vectors is found by multiplying their corresponding components and adding the results.
For two vectors and in , their dot product (written as ) is defined as:
Let's walk through an example. Suppose we have two vectors in :
To find their dot product, , we multiply the corresponding components and add them up:
The result is , a single scalar number.
An important rule, similar to vector addition, is that you can only calculate the dot product of two vectors that have the same number of components. You cannot take the dot product of a vector in and a vector in .
More Examples and Notation
Let's get some more practice with the calculation. The following video provides several clear examples and introduces a slightly different but common notation for vectors.
The video "Dot Product of Two Vectors" from The Organic Chemistry Tutor provides clear, step-by-step examples.
First, watch the examples for 2D and 3D vectors from the beginning until the end of the 3D example. The video uses the notation ax i + ay j. This is common in physics and engineering. The i, j, and k represent the standard basis vectors for the x, y, and z axes, respectively. For our purposes, you can simply think of the number in front of i as the first component, the number in front of j as the second, and so on. For example, the vector 5i - 4j + 3k is equivalent to the column vector \begin{bmatrix} 5 \\ -4 \\ 3 \end{bmatrix}.
Combining Operations
The real power of these operations comes from combining them. What happens when the dot product meets vector addition or scalar multiplication, which you learned about in the last lesson? The properties you expect, like distributivity, generally hold.
Let's see how this works in practice.
The same video from The Organic Chemistry Tutor demonstrates how to handle dot products when they are combined with other vector operations.
Please watch the next three examples in the video, starting from this point. The first example calculates \vec{a} \cdot (\vec{b} + \vec{c}). Notice how vector addition is performed first, as per the order of operations, before the dot product is taken. The second example shows how to compute (3\vec{a}) \cdot \vec{b}. Here, the scalar multiplication is done first. The final example calculates 4(\vec{a} \cdot \vec{b}). Notice the subtle but important difference: the dot product is calculated first, resulting in a scalar, which is then multiplied by another scalar.
These examples highlight an important point: the dot product behaves predictably with the other operations we've learned. Specifically, for any vectors and any scalar :
- Commutativity:
- Distributivity:
- Associativity with scalar multiplication:
You don't need to memorize these properties by name right now, but you should be comfortable with the underlying calculations, which follow the standard order of operations.
Conclusion
In this lesson, we introduced the dot product, a fundamental tool in linear algebra. You've learned how to compute it and seen how it interacts with the vector operations you already know.
The key takeaways are:
- The dot product is an operation that takes two vectors and produces a single scalar number.
- It is calculated by multiplying the corresponding components of the two vectors and then summing those products.
- The dot product can only be calculated between vectors of the same dimension.
- It follows familiar algebraic properties like commutativity and distributivity when combined with vector addition and scalar multiplication.
So far, we've treated the dot product as a purely mechanical calculation. In our next lesson, we will unlock its true power by using it to define the magnitude (or norm) of a vector. This will be our first major step in using this algebraic tool to answer fundamental geometric questions about length.