Given the points in the three-dimensional affine space $A = (1, 1, 1)$, $B = (-2, -1, 0)$.
• Find the Cartesian equations of line $r$ for points $A$, $B$.
I found it and it is:
$$
\left\{
\begin{array}{c}
x-3z+2=0 \\
y-2z+1=0 \\
\end{array}
\right.
$$
Correct me if it's wrong. Now given the line $r'$ of parametric equations:
$$
\left\{
\begin{array}{c}
x=1+3t' \\
y=-2 \\
z=1-t'
\end{array}
\right.
$$
with $t'∈R$, how can I prove that the lines $r$ and $r'$ are not parallel?
Best Answer
Hint: a direction vector of the first line is given by $A-B=(3,2,1)$, while a direction vector of the second line (look at the coefficients of $t'$) is $(3,0,-1)$.