Triangle Inequality and the Triangle Law of Vector Addition

[Ryuzaki]

The triangle inequality states that, the sum of any two sides of a triangle must be greater than the third side of the triangle.

But the triangle law of vector addition states that if we can represent two vectors as the two sides of a triangle in one order ,the third side of the triangle taken in the reverse order will be the sum of the two vectors.

However, the above two laws seem to contradict each other; one states that the third side should be less than the sum of the other two sides, while the other law states that it would be equal (considering the magnitudes of the vectors).

So can anyone tell me what I'm missing here? Thanks.

 

[oli4]

Hi Ryuzaki
The third vector is the sum of the two first vectors, but its length is not the sum of the lengths of the vectors.

 

[mathman]

The length of the third vector can be calculated using the law of cosines.

 

[Ryuzaki]

Thank you, oli4 and mathman. Seems like I had a pretty silly misconception there.

 

[CellCoree]

I can provide some clarification on the apparent contradiction between the triangle inequality and the triangle law of vector addition. Both of these laws are valid and serve different purposes in mathematics and physics.

The triangle inequality is a fundamental geometric principle that applies to all triangles, regardless of their orientation or shape. It states that the sum of any two sides of a triangle must always be greater than the third side. This is a necessary condition for a triangle to exist and is based on the properties of Euclidean geometry.

On the other hand, the triangle law of vector addition is a specific application of the triangle inequality in vector mathematics. It states that if we represent two vectors as the two sides of a triangle in a specific order, the third side of the triangle taken in the reverse order will be the sum of the two vectors. This law is based on the properties of vector addition and is applicable in physics and engineering.

It is important to note that the triangle law of vector addition only applies to vectors, which have both magnitude and direction. The triangle inequality, on the other hand, applies to all geometric shapes, including triangles, regardless of whether they represent vectors or not.

In summary, the triangle inequality and the triangle law of vector addition are not contradictory, but rather complementary laws that serve different purposes in mathematics and physics. The triangle inequality is a general geometric principle, while the triangle law of vector addition is a specific application of this principle in vector mathematics.