Set the two equations equal to each other. Then for each of the three coordinates, you will get an equation in $s$ and $t$, so you will have three equations in two variables. If the two lines are co-planer then there is a unique solution for $s$ and $t$ which will give you the coordinates of the point of intersection.
For example, the first coordinates give you the equation
$$ 2+3t=5-3s $$
Find the equations for the other two coordinates and finish the problem.
ADDENDUM:
Now that you correctly found the point of intersection $(5,0,3)$ you have the necessary information to find the equation of the plane which contains the two intersecting lines.
To find the equation of a plane containing two intersecting lines you need three pieces of information: direction vectors for each of the two lines and the point of intersection of the two lines.
The direction vectors are the vector coefficients of your two vector line equations:
- $\langle 3,-3,3\rangle$
- $\langle 3,-3,0\rangle$
These two may be simplified by multiplying by $\dfrac{1}{3}$ since multiplication by a nonzero constant does not change the direction of a vector. So use the following for the two direction vectors.
- $\langle 1,-1,1\rangle$
- $\langle 1,-1,0\rangle$
The cross-product of these two vectors gives a normal vector $N=\langle a,b,c\rangle$ for the plane containing the two lines.
$N=\begin{vmatrix}
\mathbf{i}&\phantom{-}\mathbf{j}&\mathbf{k}\\1&-1&1\\1&-1&0\end{vmatrix}=\mathbf{i}+\mathbf{j}=\langle 1,1,0\rangle$
The equation of a plane has the form
$$ ax+by+cz=d $$
where the normal vector is $\langle a,b,c\rangle$ and $d$ is computed using the values of $a,b,c$ and the coordinates of a point in the plane ( in this case, $(5,0,3)$).
So the equation for this plane is
$$ 1\cdot x+1\cdot y+0\cdot z=d$$
where
$$1(5)+1(0)+0(3)=d=5$$
So the equation of the line containing the two lines is
$$ x+y=5 $$
Best Answer
Two examples: $p_1=(0,0),q_1=(1,1),\ \ p_2=(2,0),q_2=(0,1)$ $p_1=(0,0),q_1=(1,1),\ \ p_2=(2,1),q_2=(0,-1)$ The first example will have positive values for both s's. The second example will have a negative value for $s_2$.
I'll let you do the arithmetic.