# Help Understanding Dot Product

• Opus_723
In summary, the dot product between two vectors a and b can be calculated as |a||b|cosΘ, where Θ is the angle between the two vectors. This relationship can be understood by looking at the Cauchy-Schwarz inequality and the cosine law. The dot product provides a description of the geometry of the vector space and can be used to calculate lengths and angles. The dot product can also be extended to higher dimensions and different geometries. The history of the development of the dot product and cross product can be found in sources on the history of mathematics.
Opus_723
I'm learning about dot products, and I'm having a bit of trouble grasping why axbx + ayby + azbz = ab*cosΘ. I understand how it works in two dimensions, I think, but three is still fuzzy.

This is what I came up with for two dimensions. The angle between the vectors is simply the difference between the angle between each vector and the x-axis.

ab*cosΘ
= ab*cos(Θab)
= ab*(cosΘacosΘb + sinΘasinΘb)
= a*cosΘab*cosΘb + a*sinΘab*sinΘb
= axbx + ayby

Now I understand intuitively that the dot product rule should work in three dimensions, since you could always orient the axes so that the two vectors lie on the x-y plane, in which case the above applies. But I'd like to see an analytical proof like the one above, except including azbz. I would feel a lot better about this if I could see that.

If you rotate the vectors a and b in a way such that a is aligned along the z axis, the expression for the scalar product reduces to $a \cdot b=a b_z =a b cos \theta$. The last passage follows the change of coordinate system (from cartesian to spherical).

Opus_723 said:
I'm learning about dot products, and I'm having a bit of trouble grasping why axbx + ayby + azbz = ab*cosΘ. I understand how it works in two dimensions, I think, but three is still fuzzy.

This is what I came up with for two dimensions. The angle between the vectors is simply the difference between the angle between each vector and the x-axis.

ab*cosΘ
= ab*cos(Θab)
= ab*(cosΘacosΘb + sinΘasinΘb)
= a*cosΘab*cosΘb + a*sinΘab*sinΘb
= axbx + ayby

Now I understand intuitively that the dot product rule should work in three dimensions, since you could always orient the axes so that the two vectors lie on the x-y plane, in which case the above applies. But I'd like to see an analytical proof like the one above, except including azbz. I would feel a lot better about this if I could see that.

Hey there Opus_723.

Dot products (as well as general inner products of which a dot product is one example of an inner product) provide a description of geometry of the respective vector space.

The best way to think about this is to think first about the notion of length. This is captured by what is called a norm.

One norm you should be familiar with is the pythagorean theorem of a triangle. c^2 = a^2 + b^2, where a and b are the sides of a triangle and c is the length of a hypotenuse. The sides a and b are perpendicular to each other and the length of c is given by SQRT(a^2 + b^2).

Now it turns out that in an inner product space (provided that it meets the axioms) can be written in terms of a norm. We can form the inner product using the definition of the norm, and since the norm is known with euclidean geometry (the pythagorean theorem but in any dimension), we can use this to get an expression of the inner product for euclidean spaces (think right angle geometry that you are used to).

In geometries that are not "flat" (i.e. curved geometries), we have to use more advanced mathematics, but the idea is the same. If the conditions for an inner product are satisfied, you can find ways of calculating the inner product in that geometry.

Hermann Grassman wrote a theory about geometric calculus using inner and outer products and if you read modern accounts of geometric calculus, you'll find that these come in at a very abstract level. For linear systems, these are more straightforward, but for general systems (i.e. curved space), you are dealing with differentials and it can get hairy.

Also with regard to the |a||b|cos(theta) the best way to understand this is to look at the Cauchy-Schwarz inequality. This inequality says that the inner product (or in your case dot product) |<a,b>| = |a . b| <= ||a|| x ||b|| where |x| is the absolute value of x, and ||a|| is the length of vector a. This equality establishes that the cos(theta) term relates to an "angle" and is why we associate geometry with inner products.

I use the cosine law for the intuition. Let's say you have a vector a and b. Their difference is a-b. These 3 vectors can form a triangle. Let $/theta$ be the angle between a and b. By the cosine law, we have
$$|\mathbf{a}-\mathbf{b}|^2=|\mathbf{a}|^2+|\mathbf{b}|^2-2|\mathbf{a}||\mathbf{b}|\cos\theta.$$
Since the norm of the vector squared is equal to the vector dot itself, we have
$$(\mathbf{a}-\mathbf{b})\cdot(\mathbf{a}-\mathbf{b})=\mathbf{a}\cdot \mathbf{a}+\mathbf{b}\cdot \mathbf{b} -2|\mathbf{a}||\mathbf{b}|\cos\theta.$$
Since the dot product is distributive, we have
$$\mathbf{a}\cdot \mathbf{a}-2\mathbf{a}\cdot\mathbf{b}+\mathbf{b}\cdot \mathbf{b}=\mathbf{a}\cdot \mathbf{a}+\mathbf{b}\cdot\mathbf{b} -2|\mathbf{a}||\mathbf{b}|\cos\theta.$$
Simplifying yields
$$-2\mathbf{a}\cdot \mathbf{b}=-2|\mathbf{a}||\mathbf{b}|\cos\theta.$$
Further simplifying yields
$$\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta.$$

dalcde, Thanks, that makes the most sense so far. That satisfies me enough to start using it. As a side note, does anyone know where I could find some history as to how to dot and cross products were originally developed?

## What is the dot product?

The dot product, also known as the scalar product or inner product, is a mathematical operation that takes two vectors and produces a scalar quantity. It is calculated by multiplying the corresponding components of the vectors and then summing them together.

## What is the geometric interpretation of the dot product?

The dot product has a geometric interpretation as the product of the magnitudes of two vectors and the cosine of the angle between them. This means that the dot product can be used to determine the angle between two vectors, as well as the projection of one vector onto another.

## How is the dot product related to vector multiplication?

The dot product is an operation that is defined for two vectors in any dimension, while vector multiplication is only defined for specific types of vectors (such as cross product for 3D vectors). However, the dot product is closely related to vector multiplication, as they both involve combining the components of two vectors to produce a new vector or scalar quantity.

## What is the significance of a zero dot product?

A zero dot product between two vectors means that they are perpendicular to each other, or at a 90-degree angle. This is because the cosine of 90 degrees is 0, and therefore the dot product will be 0. In other words, a zero dot product indicates that the two vectors do not have any component in the same direction.

## How is the dot product useful in physics and engineering?

The dot product has many practical applications in physics and engineering. It can be used to calculate work, power, and energy in systems with forces acting on them. It is also used in calculating torque, which is important in understanding rotational motion. In addition, the dot product is used in vector calculus to solve problems in fields such as fluid mechanics and electromagnetism.

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