Using the cosine theorem, the formula of dot product is derived from the Point product definition.

First, the cosine theorem is proved:
Having an edge of a, B, c, the corresponding angle is a_angle, B _angle, c_angle
The following relationship can be found as a diagonal line from the point to the corresponding edge:
(1)
A = B * cos c_angle + c * cos B _angle
(2)
B = a * cos c_angle + c * cos a_angle
(3)
C = a * cos B _angle + B * cos a_angle
For (1) *:
A ^ 2 = a * B * cos c_angle + a * c * cos B _angle
There are also:
B ^ 2 = B * a * cos c_angle + B * c * cos a_angle
C ^ 2 = c * a * cos B _angle + c * B * cos a_angle
Compared with the above three, we can find that:
A ^ 2 + B ^ 2 = {a * B * cos c_angle + a * c * cos B _angle} + {B * a * cos c_angle + B * c * cos a_angle}
GO
The items include:
A ^ 2 + B ^ 2 = {a * B * cos c_angle + B * a * cos c_angle} + {a * c * cos B _angle ++ B * c * cos a_angle}
GO
A ^ 2 + B ^ 2 = 2 * a * B * cos c_angle + c ^ 2
Continue to prove:
Assume that a, B, and c are in three-dimensional coordinates.
(4)
A ^ 2 = a1 ^ 2 + a2 ^ 2 + a3 ^ 2
There are also:
(5)
B ^ 2 = b1 ^ 2 + b2 ^ 2 + b3 ^ 2
(6)
C ^ 2 = (a1-b1) ^ 2 + (a2-b2) ^ 2 + (a3-b3) ^ 2
To simplify (6:
C ^ 2 = a1 ^ 2 + a2 ^ 2 + a3 ^ 2 + b1 ^ 2 + b2 ^ 2 + b3 ^ 2-2 * a1 * b1-2 * a2 * b2-2 * a3 * b3
Comprehensive (4), (5:
A ^ 2 + B ^ 2-2 * a * B * cos c_angle = a ^ 2 + B ^ 2-2 * a1 * b1-2 * a2 * b2-2 * a3 * b3
Go
2 * a * B * cos c_angle = 2 * a1 * b1 + 2 * a2 * b2 + 2 * a3 * b3
Go
A * B * cos c_angle = a1 * b1 + a2 * b2 + a3 * b3
The dot product is defined as follows: point_a * point_ B = a1 * b1 + a2 * b2 + a3 * b3
The formula point_a * point_ B = a * B * cos c_angle is obtained.

From: chenbingchenbing's column