Prove with vectors that for every triangle the lenght of one side is smaller than the sum of the other two sides.
To prove that for every triangle, the length of one side is smaller than the sum of the other two sides, we can use vector representation. Let's consider a triangle with vertices A, B, and C. We can represent each vertex as a vector: \(\overrightarrow{AB}\), \(\overrightarrow{BC}\), and \(\overrightarrow{CA}\).
To prove the statement, we need to show that the length of any one side is smaller than the sum of the lengths of the other two sides.
Using vector representation, the length of a side can be calculated as the magnitude of the vector between its endpoints. In this case, we can represent the length of side AB as \(\left \| \overrightarrow{AB} \right \|\), the length of side BC as \(\left \| \overrightarrow{BC} \right \|\), and the length of side CA as \(\left \| \overrightarrow{CA} \right \|\).
Now, let's assume a triangle with vectors \(\overrightarrow{AB}\), \(\overrightarrow{BC}\), and \(\overrightarrow{CA}\). We can express \(\overrightarrow{AB}\) in terms of \(\overrightarrow{BC}\) and \(\overrightarrow{CA}\) as follows:
\(\overrightarrow{AB} = \overrightarrow{BC} + \overrightarrow{CA}\)
Taking the magnitudes on both sides of the equation, we have:
\(\left \| \overrightarrow{AB} \right \| = \left \| \overrightarrow{BC} + \overrightarrow{CA} \right \|\)
Using the triangle inequality property for vectors, we know that:
\(\left \| \overrightarrow{BC} + \overrightarrow{CA} \right \| \leq \left \| \overrightarrow{BC} \right \| + \left \| \overrightarrow{CA} \right \|\)
Substituting this back into the equation, we have:
\(\left \| \overrightarrow{AB} \right \| \leq \left \| \overrightarrow{BC} \right \| + \left \| \overrightarrow{CA} \right \|\)
This equation demonstrates that the length of side AB is smaller than or equal to the sum of the lengths of sides BC and CA, and since we are assuming a triangle, the lengths are not equal (otherwise, the triangle would be degenerate). So, we can conclude that:
\(\left \| \overrightarrow{AB} \right \| < \left \| \overrightarrow{BC} \right \| + \left \| \overrightarrow{CA} \right \|\)
Hence, we have proven that for every triangle, the length of one side is smaller than the sum of the other two sides using vector representation and the triangle inequality for vectors.