Vector Magnitude with Dot Products

In our previous lesson, we introduced the dot product, a way of multiplying two vectors to get a scalar. We focused on the mechanics of the calculation: $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n}$ . While this might seem like a purely abstract algebraic rule, today we'll see that it has a profound geometric meaning.

This lesson bridges that gap between algebra and geometry. We will use the dot product to define and calculate one of the most fundamental properties of a vector: its length, also known as its magnitude or norm. By the end of our session, you'll be able to compute the length of any vector using the dot product, a skill that is foundational to nearly every application of linear algebra.

Defining the Length of a Vector

Before we even bring the dot product into the picture, let's think about what the length of a vector means intuitively. If you have a vector in a 2D plane, like $v = [34]$ , how would you find its length? You might already see the connection to a classic geometric tool.

The Khan Academy video below provides a clear definition of vector length, showing how it arises naturally from the Pythagorean theorem.

Vector dot product and vector length | Vectors and spaces | Linear Algebra | Khan Academy

This segment from "Vector dot product and vector length" formalizes the concept of vector length.

Watch the portion from this timestamp. Pay close attention to two ideas: The general formula for the length of a vector in \mathbb{R}^n. How this formula is a direct extension of the Pythagorean theorem you're familiar with from 2D and 3D geometry.

As the video explains, the length of a vector $a$ in $R^{n}$ is denoted by $∥ a ∥$ . It's calculated by taking the square root of the sum of the squares of its components:

∥ a ∥ = a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2}

The terms length, magnitude, and norm are used interchangeably, so you should be comfortable with all three.

The Key Connection: Length via the Dot Product

Now, let's connect this back to the dot product from our last lesson. Consider what happens when we calculate the dot product of a vector with itself. Let's take our vector $a$ again:

a \cdot a = a_{1} (a_{1}) + a_{2} (a_{2}) + \dots + a_{n} (a_{n}) = a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2}

Notice something familiar? The result, $a \cdot a$ , is exactly the quantity inside the square root in our formula for length.

This gives us a powerful and elegant new way to express the length of a vector:

∥ a ∥^{2} = a \cdot a

And by taking the square root of both sides, we get the main result for this lesson:

∥ a ∥ = a \cdot a

This is a crucial identity in linear algebra. It establishes that the dot product isn't just an arbitrary calculation; it's intrinsically linked to the concept of length.

Let's watch the next segment of the same Khan Academy video, which makes this connection explicit.

Vector dot product and vector length | Vectors and spaces | Linear Algebra | Khan Academy

The video now directly proves the relationship between the length of a vector and the dot product of the vector with itself.

Please watch from the start of this short segment. The key takeaway is the final relationship shown on screen: the length squared equals the vector dotted with itself.

Calculation and Examples

Let's put this into practice. Suppose we want to find the magnitude of the vector $v = 1 - 2 3$ .

Calculate the dot product of the vector with itself:
$v \cdot v = (1) (1) + (- 2) (- 2) + (3) (3) = 1 + 4 + 9 = 14$
Take the square root of the result:
$∥ v ∥ = v \cdot v = 14$

So, the length of the vector $v$ is $14$ . This single, elegant procedure works for a vector of any dimension.

A vector with a length of 1 is called a unit vector. Unit vectors are very important as they are often used to simply specify a direction. For example, is $u = [1/2 3 /2]$ a unit vector?

Let's check:

u \cdot u = (\frac{1}{2})^{2} + (\frac{3}{2})^{2} = \frac{1}{4} + \frac{3}{4} = 1

Since $∥ u ∥ = 1 = 1$ , yes, $u$ is a unit vector.

The Power of a Single Operation

At first glance, the formula $∥ v ∥ = v \cdot v$ might seem a bit circular. The dot product is defined using the vector's components, and so is the length. So what's the big deal?

The "big deal" is that it establishes the dot product as a fundamental building block. It shows that one single algebraic operation—the dot product—can be used to compute a critical geometric property. This is an example of the elegance and power of abstraction in mathematics.

The next video gives an excellent perspective on why this relationship is so significant. It frames the dot product as a "tool" that, once you have it, allows you to compute many other geometric quantities.

Linear Algebra 20g: The Dot Product - One of the Most Brilliant Ideas in All of Linear Algebra

This video, "The Dot Product - One of the Most Brilliant Ideas," offers a conceptual take on the importance of this operation. While the video starts with a different definition of the dot product (which we will explore later), the segment here focuses on the utility of connecting length to the dot product.

Please watch from this section. Focus on the speaker's core argument: by expressing length in terms of the dot product, we begin to see that the dot product is a foundational operation. If you were designing a system to do geometry, implementing the dot product would be one of the most powerful and efficient things you could do. (Note: The speaker uses the term "inner product," which is a more general concept that we will see later. For now, you can treat it as a synonym for "dot product.")

As the video beautifully argues, if you could only bring one tool to solve a wide range of geometric problems, the dot product would be an excellent choice.

Conclusion

In this lesson, we have forged a critical link between the algebra of vectors and their geometry. We've moved the dot product from being a mere calculation to being a tool for measuring length.

Here are the main takeaways:

The length (or magnitude, or norm) of a vector $v$ , denoted $∥ v ∥$ , is defined as the square root of the sum of the squares of its components.
This length can be computed directly using the dot product: $∥ v ∥ = v \cdot v$ .
This relationship establishes the dot product as a fundamental operation that connects algebraic manipulation to geometric measurement.

We have now used the dot product to answer the question "how long is this vector?". In our next lesson, we will use it to answer another, equally important geometric question: "what is the angle between these two vectors?". This will allow us to define concepts like orthogonality (perpendicularity) and projections, which are central to linear algebra.

Recommended resources

Defining the Length of a Vector

The Key Connection: Length via the Dot Product

Calculation and Examples

The Power of a Single Operation

Conclusion