Triangle Inequality for Vectors

Triangle inequality states that the length of each side of a triangle is smaller than or equal to the sum of the lengths of the other two sides. Thus, it is impossible to form a triangle if the sum of its two sides equals the other two sides. 

Apart from its general form, it is also written using vectors and vector lengths (norms):

If ‘x’ and ‘y’ are two vectors in ℝⁿ, then 

||x + y|| ≤ ||x|| + ||y||

Let us plot a triangle with the length of its two sides as AB = x and BC = y. The third side is written as the sum of the vectors ‘x’ and ‘y,’ that is (x + y).

Thus, the triangle is formed as shown.

Here, we observe that equality holds when either ‘x’ or ‘y’ (or both) are zero. The inequality holds strictly when the two non-zero vectors ‘x’ and ‘y’ are in the same direction.

To prove the triangle inequality, we will first prove its lemma. 

Lemma

If ‘x’ and ‘y’ are two vectors in ℝⁿ, then ||x + y||² = ||x||² + 2(x ⋅ y) + ||y||² 

Proof

We have ||x + y||² 

By the dot product properties, we get

||x + y||² = (x + y) ⋅ (x + y)

= x ⋅ x + x ⋅ y + y ⋅ x + y ⋅ y

= ||x||² + 2(x ⋅ y) + ||y||² 

Proof

Now, we will prove the triangle inequality for vectors using the above lemma and Cauchy Schwarz inequality.

Here, we take ||x + y||² 

By the lemma, we get

||x + y||² = ||x||² + 2(x ⋅ y) + ||y||² …..(i)

By Cauchy Schwarz inequality, we know |x ⋅ y| ≤ ||x|| ||y||

From (i), ||x + y||² ≤ ||x||² + 2 ||x|| ||y|| + ||y||² = (||x|| + ||y||)² 

⇒ ||x + y||² ≤ (||x|| + ||y||)² 

Thus, triangle inequality for vectors is proved.