Question

What geometric relationship must two vectors have in order for$\Vert \mathbf{u}+\mathbf{v}\Vert =\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert$ ?

Short answer

The geometric relationship between the vectors is that they are parallel to each other and are in same direction.

Step-by-step solution

Step 1:Given information

The given expression is$\Vert \mathbf{u}+\mathbf{v}\Vert =\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert$

Step 2:Simplification

$\text{Consider the non-zero vectors}\mathbf{u}\text{and}\mathbf{v}\text{.}$

$\text{The objective is to find the geometric relationship between two vectors if}\Vert \mathbf{u}+\mathbf{v}\Vert =\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert \text{.}$

$\text{To find the relationship, consider the value}\Vert \mathbf{u}+\mathbf{v}\Vert \text{.}$

$\text{The expression}\Vert \mathbf{u}+\mathbf{v}\Vert \text{gives:}$

$\Vert \mathbf{u}+\mathbf{v}{\Vert }^2=(\mathbf{u}+\mathbf{v})·(\mathbf{u}+\mathbf{v})$

$\Vert \mathbf{u}+\mathbf{v}{\Vert }^2=\Vert \mathbf{u}{\Vert }^2+\Vert \mathbf{v}{\Vert }^2+2\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert \cos \theta \dots \dots$

$\text{For}\theta =0°\text{, the equation (1) reduces to:}$

$\Vert \mathbf{u}+\mathbf{v}{\Vert }^2=\Vert \mathbf{u}{\Vert }^2+\Vert \mathbf{v}{\Vert }^2+2\Vert \mathbf{u}\Vert \mathbf{v}\Vert$

$\Vert \mathbf{u}+\mathbf{v}{\Vert }^2=(\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert {)}^2$

$\Vert \mathbf{u}+\mathbf{v}\Vert =\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert$

$\text{The result}\Vert \mathbf{u}+\mathbf{v}\Vert =\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert \text{holds when}\theta =0°\text{.}$

Therefore, the geometric relationship between the vectors is that they are parallel to each other and are in same direction.