Q .11 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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.