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# Determine properties of a point whose movement in space is determined by a vector parametric equation

Consider a point moving in space with position at time given by 1. Prove that the motion of the point is along a line.
2. Find a vector parallel to this line.
3. Find the time at which the point intersects the plane with Cartesian equation .
4. What is the Cartesian equation for the plane parallel to the plane in part (c) which contains the point ?
5. Let be the plane perpendicular to containing the point . Find a Cartesian equation for .

1. 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 2. From part (a) we have a vector parallel to given by .
3. First, the line on which the point moves is the set of points So, to find the intersection with the plane we compute 4. 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 5. 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 