Is the main purpose of the dot product to find the angle between 2 vectors?

Question

From what I have read, there is no trigonometric way to find the angle between the hypotenuse and the adjacent side of a triangle unless that triangle is a right triangle. Or alternatively said, it is trigonometrically impossible to find an angle between 2 sides of an oblique triangle, if all you know is the length of those 2 adjacent sides.

But using the dot product, we are able to find the length required to make the adjacent side perpendicular to the opposite side, where that opposite side intersects the hypotenuse. To clarify what I mean I attached an illustration: click me

Then we can use trigonometry to find the angle between the hypotenuse and the adjacent side because now this triangle is a right triangle.

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My questions is:

1. I've been trying to understand the importance of the dot product geometrically. However, I'm recently wondering if the real importance of the dot product is not what it geometrically represents but what it can be used to find. So is the real importance/purpose of the dot product is that we can use it to "convert" 2 vectors into 2 vectors that form a right triangle so we can find the angle between them knowing only the magnitude of those two vectors?

Answer 1

You are right that, given the definition of the doc product, $\vec{a} \cdot \vec{b}=ab\cos\theta$, its direct application is to find the angle between the two vectors. However, such definition has many useful properties, such as the distributive one below,

$$\vec{x}\cdot (\vec{a} - \vec{b})=\vec{x}\cdot\vec{a} -\vec{x}\cdot\vec{b} $$

which in turn allows for a lot of other applications. For example, for a triangle with vertexes $\vec{a}$, $\vec{b}$ and $\vec{c}$, we have

$$\vec{c} = \vec{a} - \vec{b}$$

One could use the dot product and its distributive property to derive the cosine rule fairly effortlessly,

$$\vec{c}\cdot\vec{c} = (\vec{a}-\vec{b})\cdot (\vec{a}-\vec{b})$$

$$c^2= a^2 -2\vec{a}\cdot \vec{b} + b^2$$

$$c^2= a^2 + b^2 -2ab\cos\theta$$

Note that the proof of the Pythagorean formula is just a special case.

The derivation of the cosine rule is a convincing example for illustrating the usefulness of the dot product. There are many other useful and important applications as well in the vector space and with Cartesian coordinates.

Answer 2

The reason you cannot find any angles of a triangle if you know only two sides is that for any two lengths there are infinitely many different triangles (with different angles), each of which has sides of those two lengths.

But if you know three sides of a triangle you can use trigonometry to find all three angles. You don’t need to construct any right triangles to do that.

To get the angle between two vectors using the dot product, again you don’t have to construct a right triangle. Making a right triangle is one useful thing you can do with a dot product, but if you just want the angle you just need to put the known quantities in the formula and solve for the angle.

But there are many, many useful things you can do with dot products when you don’t care at all about the angle between the two vectors in the dot product. You just haven’t seen these applications yet.

So I would say that finding the angle between vectors is not the only reason for the dot product. It is not even the main reason for dot products. It merely one of many reasons.

Answer 3

I am not so sure dot product is the most efficient way to get the angle

Example: Find angles between 2 vectors, A=(1,7), B=(4,6)

Using dot product, we have to do this: $\theta = \cos^{-1}\left({1*4 + 7*6 \over \sqrt{1^2+7^2} \sqrt{4^2 + 6^2}}\right)≈ 0.4461$

Now, compare to this: ∠BOA = atan2(7,1) - atan2(6,4) ≈ 0.4461

note: angles measured in counterclockwise order