Q .8. - Calculus

Question

Explain how the direction vectors of two lines in ${\mathrm{ℝ}}^3$ can be used to determine whether the two lines are parallel or identical. If the two lines are not parallel or identical, what are the possibilities for them? 

Step-by-Step Solution

Answer

If the direction vectors of the two lines are not multiples of each other, then the lines are either intersecting or skew lines.

Step 1:Given information

The direction vectors of two lines in ${\mathrm{ℝ}}^3$

Step 2:Explaination

In $3d-$ space the lines either intersect, are parallel, or are identical. There is also one more possibility that the lines may be skew lines.

For any given two line equations we use the direction vectors to know the relation between then If the direction vectors of two lines are scalar multiples of each other, then the lines are either parallel or identical.

If the direction vectors of the two lines are not multiples of each other, then the lines are either intersecting or skew lines.

Therefore, if the two lines are not parallel or identical, then the possibility is they are intersecting or skew lines.

Hence this is the explanation.