StatusX said:Do they mean the following triangle inequality:
[tex]||\vec{a}|| + ||\vec{b}|| \le ||\vec{a} + \vec{b}||[/tex]
because it can be proved from that by picking a and b correctly.
cscott said:Yeah but it should be ||a|| + ||b|| >= ||a + b||
river_rat said:Start with
[tex]||\vec{u} + \vec{v}|| - ||\vec{v}|| \le ||\vec{u}||[/tex]
and choose
[tex]\vec{u} + \vec{v} = \vec{a}[/tex]
The rest should be pretty self evident after that.
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. This is directly related to the equation ||a|| - ||b|| ≤ ||a - b|| because it shows that the difference between the lengths of two vectors (||a|| and ||b||) must be less than or equal to the length of the vector connecting the two endpoints (||a - b||).
The triangle inequality is true because it is a mathematical property that is based on the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship holds true for all triangles, not just right triangles, and is the foundation for the triangle inequality.
The triangle inequality can be applied in various real-world situations, such as in geometry and physics. In geometry, it is used to determine the shortest distance between two points and to prove the existence of triangles with certain properties. In physics, it is used to describe the relationships between forces and their components.
One example of using the triangle inequality to solve a problem is in finding the shortest distance between two points. If we have two points A and B, we can create a triangle by connecting them with a straight line. The length of this line (the hypotenuse of the triangle) must be less than the sum of the lengths of the other two sides (the distances from A to a third point and from B to the same third point). By applying the triangle inequality, we can set up an equation and solve for the shortest distance between A and B.
Some other important properties related to the triangle inequality include the reverse triangle inequality (||a - b|| ≤ ||a|| + ||b||), which states that the magnitude of the difference between two vectors is always less than or equal to the sum of their magnitudes, and the generalized triangle inequality, which extends the concept to more than two vectors (||a + b + c + ...|| ≤ ||a|| + ||b|| + ||c|| + ...).