1. Assigned work: pg. 468 #3-8,9c,10,11,13-15 Any vector perpendicular to a plane is a “normal ” to the plane. It can be found by the Cross product of any 2 direction vectors of the plane. The normal to a plane is used to determine many properties of a plane

2. 8.5 Scalar Equation of a Plane in Space Proof of Scalar Equation of a Plane: Let be 2 points on the plane. Let the normal to the plane be

3. 8.5 Scalar Equation of a Plane in Space Therefore the Cartesian (Scalar) Equation of a Plane is: Where : A>0 and A,B,C,D are integers

4. 8.5 Scalar Equation of a Plane in Space Ex 1: Find the Cartesian/Scalar equation of the plane: a) That passes through point (-3,1,-7) and has normal vector (2,4,-5)

5. 8.5 Scalar Equation of a Plane in Space Ex 1: Find the Cartesian/Scalar equation of the plane: b) that represents the xz plane.

6. 8.5 Scalar Equation of a Plane in Space Similarly equations for: xy plane: z=0 yz plane: x=0

7. 8.5 Scalar Equation of a Plane in Space c) that contains the points A(2,4,-1), B(3,0,2) and C(-1,-2,5).

8. 8.5 Scalar Equation of a Plane in Space

9. Ex 2: Find the vector and parametric equations of the plane,, parallel to : and passing through the point B(2,3,-1).

10. 8.5 Scalar Equation of a Plane in Space Therefore the vector equation is: The parametric equations are:

11. 8.5 Scalar Equation of a Plane in Space The Catesian/Scalar equation is used most often because: 1) It is simpler than the vector or parametric forms. 2) Unlike vector or parametric forms, there is only ONE Cartesian/Scalar equation for each plane. The parametric equations are:

12. 8.5 Scalar Equation of a Plane in Space From a Scalar equation we can easily identify the normal. The normal is often used to: ***Identify whether two planes are parallel, coincident or perpendicular. The parametric equations are:

13. 8.5 Scalar Equation of a Plane in Space Coincident Planes: - Scalar equations of planes are scalar multiples of each other. Ex:

14. 8.5 Scalar Equation of a Plane in Space Parallel Planes: - When normal vectors are parallel and they don’t share a common point. Ex: Perpendicular Planes: -When normal vectors are perpendicular. Ex:

15. 8.5 Scalar Equation of a Plane in Space Ex 3: Determine whether the following planes are parallel, coincident, perpendicular or neither. a) b)

16. 8.5 Scalar Equation of a Plane in Space Normals can also be used to find whether a line is parallel and off, parallel and on or perpendicular to a plane. Line Perpendicular to a Plane: Line Parallel to a Plane: (parallel and on if they ALSO share a common point)

17. 8.5 Scalar Equation of a Plane in Space Ex. 4: Is the line parallel to the plane ? Is so, does it lie on or off the plane?

18. 8.5 Scalar Equation of a Plane in Space But we must now check if the line is on or off the plane.

19. 8.5 Scalar Equation of a Plane in Space Therefore line is parallel and off the plane.

20. 8.5 Scalar Equation of a Plane in Space Angle between two planes with normals