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Last modified on May 25th, 2024

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 ℝ^{n}, 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.

If ‘x’ and ‘y’ are two vectors in ℝ^{n}, 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}

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.

Last modified on May 25th, 2024