Question

Векторы a,b,c связаны условием a+b+c=0 , доказать,что a×b=b×c=c×a. Каков геометрический смысл этого результата?

Answer 1

Из условия выразим переменную а, тоесть a=-b-c, 
  Подставим вместо а.
(a×a=b×b=c×c = 0)
Свойство: при перестановки b×c знак меняется на противоположный -c×b(-b-c)×b = -b×b - c× b = -c×b=b×c 

  Доказали что a×b=b×c
Аналогично докажем что a×b=c×a и b×c=c×a
Из уравнения выразим переменную b : b=-a-c
 b×c = (-a-c)×c = -a×c - c×c = -a×c = c×a - ДОКАЗАЛИ
Из уравнения выразим переменную b: b=-a-c
 a×b =a× (-a-c)=a×(-a) - a×c = -a×c = c×a - ДОКАЗАЛИ

Answer 2

Given Condition

The given condition is that a + b + c = 0.

Proof of a×b=b×c=c×a

To prove that a×b = b×c = c×a, we can use the fact that the sum of vectors is zero.

Let's start with the cross product of a and b: a×b = -(b×a) Now, let's consider the cross product of b and c: b×c = -(c×b) Finally, let's look at the cross product of c and a: c×a = -(a×c) Since a + b + c = 0, we can rearrange the terms to get a = -b - c. Substituting this into the cross products, we get: a×b = -(-b - c)×b = b×c b×c = -(c)×c = c×a c×a = -(a)×a = a×b

Therefore, we have shown that a×b = b×c = c×a.

Geometric Interpretation

The geometric interpretation of this result is that the vectors a, b, and c form a closed loop. When the sum of vectors is zero, it means that the vectors form a closed polygon or a closed loop. In this case, the vectors a, b, and c form a closed loop, and their cross products being equal further illustrate the symmetry of this closed loop.

This result is related to the concept of cyclic order in geometry, where the order of the vectors matters in determining the direction of the cross product. The fact that a×b = b×c = c×a indicates a cyclic symmetry in the relationship between these vectors.

In summary, the result a×b = b×c = c×a signifies the geometric symmetry and the closed-loop nature of the vectors a, b, and c when their sum is zero.