Triple product formula of vector and its proof (with Python code)

The triple product formula of a vector is an identity that is often used in vector algebra , Its expression is as follows ：$$\vec{a}\times \left(\vec{b}\times \vec{c}\right)=\left(\vec{a}\cdot \vec{c}\right)\vec{b}-\left(\vec{a}\cdot \vec{b}\right)\vec{c}$$

We can use the following python Code to prove （ Let's change the above expression a little and prove this identity ）$$\vec{a}\times \left(\vec{b}\times \vec{c}\right)-\left(\vec{a}\cdot \vec{c}\right)\vec{b}+\left(\vec{a}\cdot \vec{b}\right)\vec{c}=0$$

import sympy as sym
from sympy import sin,cos,diff
x_a,y_a,z_a,x_b,y_b,z_b,x_c,y_c,z_c = sym.symbols('x_a,y_a,z_a,x_b,y_b,z_b,x_c,y_c,z_c')
a = sym.Matrix([x_a,y_a,z_a])
b = sym.Matrix([x_b,y_b,z_b])
c = sym.Matrix([x_c,y_c,z_c])
A = a.cross(b.cross(c))
B = a.dot(c)*b
C = a.dot(b)*c
print(sym.simplify(A-B+C))

The output result is ：

so , This identity holds .
Of course , We can also calculate... Manually , The amount of calculation is a little larger , It takes patience and care .

Using the triple product formula , We can get another interesting identity , namely ：
$$\vec{a}\times \left(\vec{b}\times \vec{c}\right)+\vec{b}\times \left(\vec{c}\times \vec{a}\right)+\vec{c}\times \left(\vec{a}\times \vec{b}\right)=0$$

We can also use python Procedures to prove ：

import sympy as sym
from sympy import sin,cos,diff
x_a,y_a,z_a,x_b,y_b,z_b,x_c,y_c,z_c = sym.symbols('x_a,y_a,z_a,x_b,y_b,z_b,x_c,y_c,z_c')
a = sym.Matrix([x_a,y_a,z_a])
b = sym.Matrix([x_b,y_b,z_b])
c = sym.Matrix([x_c,y_c,z_c])
A = a.cross(b.cross(c))
B = b.cross(c.cross(a))
C = c.cross(a.cross(b))
print(sym.simplify(A+B+C))

The output result is ：

so , This identity holds .
Of course , We can also calculate... Manually , Expand directly by using the triple product formula, and then merge the similar terms .