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Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below.Triangle Inequality

By considering $u$ and $v$ as two sides of a triangle, we can see that the lengths of the third side is $\| u + v \|$ and we have $\| u + v \| \leq \|u\| + \|v\|$. Under what circumstance equality occurs and how can one prove that?

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This typically follows from Cauchy-Schwartz. Try proving that first. – AnonymousCoward Dec 13 '11 at 18:09
There is a proof here: – AnonymousCoward Dec 13 '11 at 18:10
To see when you have equality, play around with the vectors in your drawing.. – AnonymousCoward Dec 13 '11 at 18:11
up vote 17 down vote accepted

I have noticed that the answer has been written down in the comments. Just to have an answer I am writing this one down.

Consider $\|u+v\|^2=(u+v) \cdot (u+v)$ where $u \cdot v$ represents the standard inner product/scalar product.Therefore $$\|u+v\|^2=\|u\|^2+2 (u \cdot v) + \|v\|^2 .$$

By the Cauchy-Schwarz Inequality we have $$u \cdot v \leq \|u\| \cdot \|v\|.$$

So, $$\|u+v \|^2= \|u\|^2+2(u \cdot v)+ \|v \|^2 \leq \|u\|^2+ 2 \|u\| \cdot \|v\| + \|v\|^2=(\|u\|+ \|v\|)^2 ,$$ i.e., $$\|u+v\|^2 \leq (\|u\|+ \|v\|)^2 \implies \|u+v\| \leq \|u \|+ \|v\| .$$

The Cauchy-Schwarz Inequality holds for any inner Product, so the triangle inequality holds irrespective of how you define the norm of the vector to be, i.e., the way you define scalar product in that vector space.

In this case, the equality holds when vectors are parallel i.e, $u=kv$, $k \in \mathbb{R}^+$ because $u \cdot v= \|u \| \cdot \|v\| \cos \theta$ when $\cos \theta=1$, the equality of the Cauchy-Schwarz inequality holds.

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Yes i understood now.Thank you all. – alok Dec 14 '11 at 6:42
@alok You are welcome. – Ramana Venkata Dec 14 '11 at 8:54

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