# Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If th

Zoe Oneal 2021-08-20 Answered
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.
$$\displaystyle{L}{1}:{\frac{{{x}}}{{{1}}}}={\frac{{{y}-{1}}}{{-{1}}}}={\frac{{{z}-{2}}}{{{3}}}}$$
$$\displaystyle{L}{2}:{\frac{{{x}-{2}}}{{{2}}}}={\frac{{{y}-{3}}}{{-{2}}}}={\frac{{{z}}}{{{7}}}}$$

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odgovoreh
The direction vectors are the numbers in the denominators of the symmetric equations.
$$\displaystyle{L}_{{1}}:{<}{1},-{1},{3}{>}$$
$$\displaystyle{L}_{{2}}:{<}{2},-{2},{7}{>}$$
One vector is not a multiple of the other. 2 times 1 is 2, but 2 times 3 isn't 7. The lines are not parallel.
Parametric form of $$\displaystyle{L}_{{1}}$$
$$\displaystyle{\frac{{{x}}}{{{1}}}}={t}\Rightarrow{x}={t}$$
$$\displaystyle{\frac{{{y}-{1}}}{{-{1}}}}={t}\Rightarrow{y}={1}-{t}$$
$$\displaystyle{\frac{{{z}-{2}}}{{{3}}}}={t}\Rightarrow{z}={2}+{3}{t}$$
Parametric form of $$\displaystyle{L}_{{2}}$$
$$\displaystyle{\frac{{{x}-{2}}}{{{2}}}}={s}\Rightarrow{x}={2}+{2}{s}$$
$$\displaystyle{\frac{{{y}-{3}}}{{-{2}}}}={s}\Rightarrow{y}={3}-{2}{s}$$
$$\displaystyle{\frac{{{z}}}{{{7}}}}={s}\Rightarrow{z}={7}{s}$$
See if the lines intersect by setting the component parts equal to each other and seeing if there is a solution.
$$\displaystyle{t}={2}+{2}{s}$$
$$\displaystyle{1}-{t}={3}-{2}{s}$$
$$\displaystyle{2}+{3}{t}={7}{s}$$
Substitute for t
$$\displaystyle{1}-{\left({2}+{2}{s}\right)}={3}-{2}{s}$$
$$\displaystyle{1}-{2}-{2}{s}={3}-{2}{s}$$
$$\displaystyle-{1}={3}$$
This is a contradiction, so there is not solution. The lines do not intersect so they are skew.