Orthogonality and the Dot Product

In our last lesson, we discovered the powerful connection between the dot product and the angle between two vectors, encapsulated in the formula $θ = arccos (\frac{a \cdot b}{∥ a ∥∥ b ∥})$ . We concluded by looking at the special case where the dot product is zero, which implies the angle between the vectors is $9 0^{\circ}$ .

This lesson is dedicated entirely to that special case. We will formally define and explore the concept of orthogonality, the precise term in linear algebra for perpendicularity. By mastering this concept, you will gain a simple yet profound tool for analyzing the geometric relationship between vectors using only a quick algebraic calculation. This is the final topic in our module on the foundations of vectors, and it lays essential groundwork for many concepts to come.

What Does "Orthogonal" Mean?

In geometry, you learned about perpendicular lines. In linear algebra, we use the term orthogonal to describe vectors that are at a $9 0^{\circ}$ angle to each other. While the words are different, the core geometric idea is the same.

The video below offers a quick explanation of the term.

Orthogonal, parallel or neither (vectors) (KristaKingMath)

This short clip from KristaKingMath introduces the concept of orthogonal vectors.

Watch from the beginning to understand how "orthogonal" relates to the more familiar term "perpendicular."

As the video explains, orthogonality signifies a 90-degree angle between two vectors.

The Dot Product Test for Orthogonality

In the previous lesson, we saw that if $a \cdot b = 0$ for two non-zero vectors, then $cos (θ) = 0$ , which means $θ = 9 0^{\circ}$ . This relationship works both ways. If the angle is $9 0^{\circ}$ , then $cos (9 0^{\circ}) = 0$ , which forces the dot product to be zero.

This gives us a simple, powerful, and definitive test for orthogonality.

Two non-zero vectors $a$ and $b$ are orthogonal if and only if their dot product is zero.
$a ⊥ b ⟺ a \cdot b = 0$

(The symbol $⊥$ is often used to denote orthogonality).

This is a cornerstone concept. Instead of calculating magnitudes and using inverse cosine to find an angle, we can perform one simple calculation—the dot product. If the result is zero, we know instantly that the vectors are orthogonal.

The video from our previous lesson summarizes this point perfectly.

The Vector Dot Product

Let's revisit the concluding insight from the "The Vector Dot Product" video by Professor Dave Explains.

Watch the brief segment from this timestamp. It clearly states that a dot product of zero is the key indicator of orthogonality.

Applying the Orthogonality Test: Examples

Let's see this test in action. To determine if two vectors are orthogonal, you simply compute their dot product.

The following video provides several clear examples of applying this test. We'll work through them one by one.

Orthogonal, parallel or neither (vectors) (KristaKingMath)

This video from KristaKingMath, "Orthogonal, parallel or neither," demonstrates the dot product test with several vector pairs.

We will focus on the parts about orthogonality. First, watch this example with two 3D vectors from the start of the problem.

In the example you just watched, the vectors were $a = ⟨ - 5, 3, 7 ⟩$ and $b = ⟨ 6, - 8, 2 ⟩$ . The dot product was calculated as:

a \cdot b = (- 5) (6) + (3) (- 8) + (7) (2) = - 30 - 24 + 14 = - 40

Since the dot product is $- 40$ , which is not zero, the vectors are not orthogonal.

Now, let's look at a case where the vectors are orthogonal.

Continue with the same video, watching the next example with 2D vectors from this point.

Here, the vectors were $a = ⟨ 4, 6 ⟩$ and $b = ⟨ - 3, 2 ⟩$ . The calculation was:

a \cdot b = (4) (- 3) + (6) (2) = - 12 + 12 = 0

Because the dot product is exactly zero, we can conclude that these two vectors are orthogonal.

Finally, the same principle applies regardless of how the vectors are written. Vectors are often expressed using the standard basis vectors $^$ , $^$ , and $k^$ , which represent the directions of the x, y, and z axes, respectively. To apply the test, you first convert this notation into the component form we're used to.

Watch the final example from the video, starting at this timestamp. Pay attention to how the vectors are first converted to component form.

The vectors were $a = - ^+ 2 ^+ 5 k^$ and $b = 3 ^+ 4 ^- k^$ .

Convert to component form:
- $a = ⟨ - 1, 2, 5 ⟩$
- $b = ⟨ 3, 4, - 1 ⟩$
Calculate the dot product: $a \cdot b = (- 1) (3) + (2) (4) + (5) (- 1) = - 3 + 8 - 5 = 0$

Again, the dot product is zero, so the vectors are orthogonal.

A Special Case: The Zero Vector

What about the zero vector, $0 = ⟨ 0, 0, \dots, 0 ⟩$ ? The dot product of the zero vector with any other vector $v = ⟨ v_{1}, v_{2}, \dots, v_{n} ⟩$ is:

0 \cdot v = (0) (v_{1}) + (0) (v_{2}) + \dots + (0) (v_{n}) = 0

The dot product is always zero. Geometrically, the concept of an "angle" with the zero vector is undefined. However, based on our algebraic test ( $a \cdot b = 0$ ), we adopt a very useful convention:

The zero vector $0$ is considered orthogonal to every vector in its space.

Conclusion

This lesson concludes our first module by focusing on one of the most elegant and useful consequences of the dot product. We have established a simple and infallible test for a fundamental geometric property.

Here are the key takeaways:

Orthogonality is the term for two vectors being perpendicular (at a $9 0^{\circ}$ angle).
The definitive test for orthogonality is the dot product: two non-zero vectors are orthogonal if and only if their dot product equals zero.
The zero vector is, by convention, considered orthogonal to all vectors.

You have now built a solid foundation in the world of vectors. You can represent them, perform basic arithmetic with them, measure their length, and determine the angle between them, including the special case of orthogonality.

In the next module, we will move from individual vectors to a more powerful structure: the matrix. A matrix can be seen as a collection of vectors, and understanding them will open the door to solving systems of linear equations, describing complex transformations, and much more. The vector properties you've learned here, especially orthogonality, will be crucial as we proceed.

Recommended resources