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Well, I was reading Apostol and in it's historical introduction section of Vector Algebra chapter he said that best features of quaternion analysis and Cartesian geometry were united and a new subject called vector algebra sprang into being. And after two lines, he said that there are applications of vector algebra in analytic geometry which are discussed in next chapter.

So, my question is how can vectors have applications into analytic geometry when the former one was created through the elements of the later one? I've tried to have a glance on the next chapter but I couldn't get to understand how did Cartesian geometry gave birth to vector algebra since it isn't mentioned in the book.

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I will not give here a general point of view. Instead, I give an example of the vector approach, here with the "dot-product tool" (in other cases, it will be the cross-product or the determinant that play a central role):

Let us look for the (common) angle $A_iOA_j$ $i \neq j$ in a regular tetrahedron $A_1A_2A_3A_4$ with centroid $O$ (an angle that every chemist knows because it is linked to methane molecule $CH_4$ (https://en.wikipedia.org/wiki/Methane)).

Instead of rather complicated considerations, it suffices to proceed like this:

$$\tag{1}\vec{OA_1}+\vec{OA_2}+\vec{OA_3}+\vec{OA_4}=0 \ \ \ \ \text{(by symmetry.)}$$

Let us assume that $\|\vec{OA_k}\|=1$ (WLOG because a scale change does not modify angles).

Then, taking the dot product of (1) with itself:

$$\sum_{k=1}^4 \vec{OA_1}^2+\sum_{i \neq j} \vec{OA_i}.\vec{OA_j}=0$$

$$4+\sum_{i \neq j} \cos(\theta)=0 \ \ \ \iff \ \ \ 4+12 \cos(\theta)=0,$$

giving $\theta=acos(-1/3) \approx 1.91$ radians $= 109.5°.$

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