# Math Help - Linear Algebra understanding question

1. ## Linear Algebra understanding question

given that $\vec{v}\in \mathbb{R}^4$
and meets the requirements:
$\vec{v}\cdot \vec{e_{2}}<0 ,$
$(\vec{e}_2+\vec{e}_4)\cdot \vec{v}=0$
$\vec{e}_1 \cdot \vec{v}=\vec{e}_4\cdot \vec{v}$
$(\vec{e}_2+\vec{e}_3)\cdot \vec{v}=0$
$\left \| \vec{v} \right \| = 6$

Find the vector $\vec{v}$ (If exists)

Thank you!

2. If we write the vector as $(v_1,v_2,v_3,v_4)$, the 5 conditions can be written as follows:

$v_2<0$

$v_2+v_4=0$

$v_1=v_4$

$v_2+v_3=0$

$v_1^2+v_2^2+v_3^2+v_4^2=36$

Letting $v_4=x$, the first four conditions give us that $v_1=v_3=v_4=x$ and $v_2=-x$ where $x>0$

By the last condition, $4x^2=36$ so that $x=\pm 3$

We reject the negative because $x$ must be positive

So the vector is $(3,-3,3,3)$

3. Thank you!
what the meaning of $\vec{e}$, Unit vector?
4. Yes. For example $\vec{e_1}=(1,0,0,0)$