Two vectors in space will either be parallel, intersecting or skew .

1. Vectors in space Two vectors in space will either be parallel,intersecting or skew. Parallel lines Lines in space can be parallel with an angle between them of 0o. Lines that meet in space will intersect each other. The angle at the point of intersection is Intersecting lines Lines that do not meet each other in space, but are not parallel, are called skew. Skew lines These lines can still have an angle between them.

2. Parallel 3d lines The parametric equations of two lines in space are given as: Show that the two lines are parallel. Turn the parametric equations into vector equations: Take the vector parts of these equations and note that one is a multiple of the other: b is -3a therefore the lines are parallel.

3. Intersecting 3d lines Show that the following 2 straight lines intersect and find the angle between them. This equation works for the x and y parts of the lines. Check it also works for the z part. The coordinate at the point of intersection is (4,5,-2). If the lines intersect each other there must be coordinate (x,y,z) at the point of intersection. To find the angle between them use the formula: So the task is to find values of s and t that satisfy all the equations. Write out the vector parts of the lines: Set up two simultaneous equations: Solving these equations gives: s=2 and t=-1

4. Skew 3d lines As the simultaneous equations do not satisfy all 3 coordinates there is no point of intersection and therefore the lines are skew. Show that the following 2 straight lines are skew and find the angle made between them. To find the angle between them use the formula: As before, set up two simultaneous equations and solve them. Write out the vector parts of the lines: from the x and y parts of the line. Solving these will give t=5 and u=-7. Now check this works for the z part of the line: Not equal.