1. Planes in Space

2. z
Equation of a Plane y x

3. z
Equation of a Plane y x

4. z
Equation of a Plane y x

5. z
Equation of a Plane y x

6. z
Equation of a Plane y b x

7. z
Equation of a Plane c y x

8. z
Equation of a Plane y x

9. z
Equation of a Plane n P y x

10. z Equation of a Plane y x P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y x

11. z Equation of a Plane y Vector Equation x Scalar Equation P(x0,y0,z0)
Q(x,y,z)
n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y
Vector Equation x
Scalar Equation

12. z Equation of a Plane y Vector Equation x Scalar Equation P(x0,y0,z0)
Q(x,y,z)
n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y
Vector Equation x
Scalar Equation

13. z Equation of a Plane y Vector Equation x Scalar Equation P(x0,y0,z0)
Q(x,y,z)
n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y
Vector Equation x
Scalar Equation

14. Examples
Find the equation of the plane through (1,1,2), (3,2,-1) and (4,2,-1).
Find the equation of the plane through (2,-1,3) and parallel to 3x – y + 4z =12.

15. Intercepts of a Plane Find the intercepts of the planes
2x – 3y + z = 6
4y + 2x = 8
z = 3
Sketch the planes.
Find the normals to the planes.

16. Examples z
Find the equation of
the plane pictured. 4 y 5 3 x

17. Graphing Planes Sketch the following planes: (a) 3x - 2y + z = 6
(b) z + 2y = 4
(c) y = 2

18. Angle Between Planes Find the angle between the two planes
2x – 3y + 4z = 6
and
x + 2y – 3z = -1