Question

Determine if the following lines are parallel, skew or intersecting.

⟨x,y,z⟩=⟨2,1,2⟩+⟨3,5,2⟩t and

⟨x,y,z⟩=⟨2,5,2 ⟩+⟨3,3,2⟩s

*If parallel (type the number 1).

*If intersecting (type the number 2).

*If skew (type the number 3).

Answer 1

$<x,y,z>=<2,1,2>+<3,5,2>t\\ <x,y,z>=<2,5,2>+<3,3,2>s\\ 1. \vec{a_1}=(3,5,2)\\ \vec{a_2}=(3,3,2)\\ \frac{3}{3}\neq\frac{5}{3}\neq\frac{2}{2}$

lines are not parallel

2.

$A(2,1,2), B(2,5,2)\\ \overrightarrow{AB}=(2-2,5-1,2-2)=(0,4,0)\\ \overrightarrow{AB}\cdot\vec{a_1}\cdot\vec{a_2}=\\ =\begin{vmatrix} 0 & 4&0 \\ 3 & 5&2\\ 3&3&2 \end{vmatrix}=0\cdot5\cdot2+4\cdot2\cdot3+0\cdot3\cdot3-\\ -0\cdot5\cdot3-3\cdot4\cdot2-3\cdot2\cdot0=24-24=0$

lines are intersecting

*If intersecting (type the number 2).