Lines and Planes in 3D Space
- The parametric equation of a line is not unique i.e. different parametric equations can represent the same line. For example, you can choose another point on the line or another direction vector that is a scalar multiple of .
- If are all non-zero, we can eliminate the parameter and write the equation of line in the following symmetric form: .
Exercise: Find the parametric equation of the line that passes through and .
Exercise: Find the plane that passes through and is parallel to the plane .
Exercise: Find the plane that passes through , , and . (Hint: Use the cross product to find a normal vector of the plane.)
Exercise: Find the line of intersection of the planes and . (Hint: The direction vector of the line of intersection is orthogonal to the normal vectors of the two planes.)
Exercise: Given the planes and . (a) Show that these two planes are parallel. (b) Find the distance between these two planes (Hint: Find a point on one of the planes and use the distance formula.)
Exercise: Let be the line passing through with the direction vector and be the line passing through with the direction vector . Find the distance between and . (Hint: Find the plane containing with the normal vector orthogonal to both and and then use the distance formula.)