Consider a point moving in space with position at time given by
- Prove that the motion of the point is along a line.
- Find a vector parallel to this line.
- Find the time at which the point intersects the plane with Cartesian equation .
- What is the Cartesian equation for the plane parallel to the plane in part (c) which contains the point ?
- Let be the plane perpendicular to containing the point . Find a Cartesian equation for .
- Proof. We use the formula for the motion of the particle to compute
This is the parametric equation for the line through parallel to the vector
- From part (a) we have a vector parallel to given by .
- First, the line on which the point moves is the set of points
So, to find the intersection with the plane we compute
- First, we have
Since we know the plane is parallel to the one in part (c) it has a Cartesian equation of the form
We compute . Hence, the plane has Cartesian equation
- Since the plane is perpendicular to the line we know that it has a normal vector in the same direction as , so (from part (b)). Thus, we have a Cartesian equation of the form
Since the point
is on the plane we have . Therefore, the plane is given by