Chapter 13

1. Chapter 13 Section 13.6 Planes

2. Equations of Planes A normal vector N for a plane is a vector that is orthogonal to every vector in the plane. This tells how the plane is oriented, it is like the “slope“ of the plane. To get the equation we form the vector from the known point on the plane to the variable point and dot it with N. The result must be zero. z y x Plane with normal and point is: Example A plane is perpendicular to the line given to the right and intersects the line at a point with x-coordinate 4. Find the equation of the plane. To find a point on the line set the x-coordinate equal to 4 and solve for t. Plug that value into l to find the point. The direction vector l can be the normal vector N for the plane. Equation:

3. Example Remember that 3 points determine a plane. Find the equation of the plane that contains the points , and . We can use any of the points P,Q, or R as a point on the plane. To find a normal vector N form the vectors in the plane and . A normal vector will be perpendicular to both of them, so take the cross product and let . We use the point to be the point on the plane and substitute in the formula. Equation of plane: Check at each point:

4. Example Find the equation of the line of intersection between the two planes whose equations are given to the right. One approach is to find two points on the line and use them to determine the direction vector d for the line. In order to do this we need to solve a system of equations and get 2 particular solutions. For these values of x and ywe substitute back into the first equation to find and in the solution. This gives the following points: The direction vector is: Equation: Solve first equation for z. Substitute: Find 2 points on this line. Check: