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Given that $(a \times b) \cdot c=2$, now I need to find the value of $[(a+b) \times(b+c)] \cdot(c+a)$.

$Assumptions = (a | b | c) ∈ Vectors[3];
Cross[a + b, b + c].(c + a) // ExpandAll
Cross[a + b, b + c].(c + a) // TensorReduce
Solve[Cross[a, b].c == 2 && 
  s == Cross[a + b, b + c].(c + a) && (a | b | c) ∈ 
   Vectors[3], s, {a, b, c}]

However, the above code cannot expand or simplify this formula according to the operation rules.

a = {x1, y1, z1};
b = {x2, y2, z2};
c = {x3, y3, z3};
Eliminate[{f == Cross[a + b, b + c].(c + a), Cross[a, b].c == 2}, {a, 
  b, c, x1, y1, z1, x2, y2, z2, x3, y3, z3}]

I want to know what I can do to simplify this formula like the reference answer and find its value according to the known conditions.

Reference answer:

$$\begin{array}{l} {[(a+b) \times(b+c)] \cdot(c+a)} \\ =[(a+b) \times b] \cdot(c+a)+[(a+b) \times c] \cdot(c+a) \\ =(a+b) \times c+(b \times c) \cdot a \\ =(a \times b) \cdot c+(a \times b) \cdot c=4 \end{array}$$

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1 Answer 1

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$Assumptions = (a | b | c) ∈ Vectors[3]
    Cross[a + b, b + c].(c + a) // TensorReduce

(* Simplify[Cross[a + b, b + c].(c + a) // TensorReduce, 
 Assumptions -> a\[Cross]b.c == 2] *)
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