# Question #c976d

Aug 26, 2016

See below

#### Explanation:

Given two vectors $\vec{a} , \vec{b}$ their sum is

$\vec{s} = \vec{a} + \vec{b}$. Now computing the norm of $\vec{s}$ and squaring

${\left\lVert \vec{s} \right\rVert}^{2} = {\left\lVert \vec{a} + \vec{b} \right\rVert}^{2} = {\left\lVert \vec{a} \right\rVert}^{2} + 2 \left\langle\vec{a} , \vec{b}\right\rangle + {\left\lVert \vec{b} \right\rVert}^{2}$

The scalar product

$\left\langle\vec{a} , \vec{b}\right\rangle = \left\lVert \vec{a} \right\rVert \left\lVert \vec{b} \right\rVert \cos \left(\hat{\vec{a} , \vec{b}}\right)$ has a maximum when $\cos \left(\hat{\vec{a} , \vec{b}}\right) = 1$ and a minimum when $\cos \left(\hat{\vec{a} , \vec{b}}\right) = - 1$ so

$\min {\left\lVert \vec{s} \right\rVert}^{2} = {\left\lVert \vec{a} \right\rVert}^{2} + {\left\lVert \vec{b} \right\rVert}^{2} - 2 \left\lVert \vec{a} \right\rVert \left\lVert \vec{b} \right\rVert$ and

$\max {\left\lVert \vec{s} \right\rVert}^{2} = {\left\lVert \vec{a} \right\rVert}^{2} + {\left\lVert \vec{b} \right\rVert}^{2} + 2 \left\lVert \vec{a} \right\rVert \left\lVert \vec{b} \right\rVert$

Finally, when two vectors $\vec{a} , \vec{b}$ are aligned, their sum is a minimum or a maximum deppending on their relative orientation:

concordant a maximum
discordant a minimum