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.
My questions is:
- 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?
Best Answer
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.