How to prove that two lines are not parallel?

Question

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?

Answer 1

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 $(3,0,-1)$.

Answer 2

In order to find the direction vector $\vec v$, you need to consider the parametric equations $P_0+\vec vt$ for the two lines

• parametric equation for line $r$ is: $$A+(A-B)t=(1,1,1)+t(3,2,1)$$
• parametric equation for line $r'$ is: $$(1,-2,1)+t'(3,0,-1)$$

Since $v_r=(3,2,1) \not \parallel v_r'=(3,0,-1)$ the two lines are not parallel.