Vectors $v_1$ and $v_2$ are given below
$v_1 =\begin{bmatrix} −1 & 2 & 3 & 0 \end{bmatrix}$
$v_2=\begin{bmatrix} −1 & 1 & −1 & 0 \end{bmatrix}$
(a) Find a non-zero vector $v_3$ which is orthogonal to both $v_1$ and $v_2$.
(b) Find a non-zero vector $v_4$ which is orthogonal to $v_1,v_2$ and $v_3$.
Hint 1
In $\mathbb{R}^3$ the vector orthogonal to the two given vectors can be obtained with cross-product: $$\vec{u}=\vec{v}\times\vec{w}=\left|\begin{array}c v_1&v_2&v_3 \\w_1&w_2&w_3 \\ \vec{i} & \vec{j} & \vec{k} \end{array} \right|$$ where $\vec{i}=(1,0,0)$, $\vec{j}=(0,1,0)$, $\vec{k}=(0,0,1)$
Hint 2
What is the dot product of $\vec{v}=(v_1,v_2,v_3,0)$ and $\vec{x}=(0,0,0,x_4)$?