1. 55: The Vector Equation of a Plane © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

2. The Vector Equation of a Plane Module C4 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" MEI/OCR

3. The Vector Equation of a Plane There are 3 forms of the equation of a plane. We are going to look at 2 of them. Suppose we have a vector n through a point A. There is only one plane through A that is perpendicular to the vector. A x n

4. The Vector Equation of a Plane There are 3 forms of the equation of a plane. We are going to look at 2 of them. Suppose we have a vector n through a point A. A x n R x Then, This is the equation of the plane since it is satisfied by the position vector of any point on the plane, including A. Suppose R is any point on the plane ( other than A ). The scalar product can be expanded to give

5. The Vector Equation of a Plane It is useful in some problems to know that a vector n will be perpendicular to the plane if it is perpendicular to 2 non-parallel vectors in the plane. C x A x There are an infinite number of vectors perpendicular to AC. For example, one lies on the plane.

6. The Vector Equation of a Plane It is useful in some problems to know that a vector n will be perpendicular to the plane if it is perpendicular to 2 non-parallel vectors in the plane. C x Others lie at angles to the plane. A x There are an infinite number of vectors perpendicular to AC. For example, one lies on the plane. Only one is also perpendicular to AB.

7. The Vector Equation of a Plane It is useful in some problems to know that a vector n will be perpendicular to the plane if it is perpendicular to 2 non-parallel vectors in the plane. C x Others lie at angles to the plane. B x There are an infinite number of vectors perpendicular to AC. For example, one lies on the plane. This one is perpendicular to the plane. Only one is also perpendicular to AB. A x

8. The Vector Equation of a Plane e.g.1 Find the equation of the plane through the point A(2, 3,  1) perpendicular to. Solution:

9. The Vector Equation of a Plane Calculating the left-hand scalar product gives the Cartesian form of the equation.

10. The Vector Equation of a Plane e.g.2 Show that the vector n is perpendicular to the plane containing the points A, B and C where Solution: The plane containing A, B and C also contains the vectors and

11. The Vector Equation of a Plane So, n is perpendicular to 2 vectors in the plane so is perpendicular to the plane.

12. The Vector Equation of a Plane  The vector equation of a plane is given by SUMMARY where a is the position vector of a fixed point on the plane n is a vector perpendicular to the plane and r is the position vector of any point on the plane. n is called the normal vector or  The Cartesian form is where n 1, n 2 and n 3 are the components of n and

13. The Vector Equation of a Plane Exercise 1. Find a vector equation of the plane through the point with normal vector 2. Find the Cartesian equation of the plane through the point perpendicular to the vector

14. The Vector Equation of a Plane 1. Plane through the point with normal vector Solution:

15. The Vector Equation of a Plane 2. Find the Cartesian equation of the plane through the point perpendicular to the vector Solution:

16. The Vector Equation of a Plane Solution: 3. Show that is perpendicular to the plane containing the points A(1, 0, 2 ), B(2, 3,  1) and C(2, 2,  1 ). n is perpendicular to 2 vectors in the plane so is perpendicular to the plane. Exercise

17. The Intersection of a Line and a Plane If a line is not parallel to a plane, it will intersect it. e.g. Find the point of intersection of the line L and the plane  given by: Solution: The point of intersection is the point with vector r satisfying both equations. Can you see how to find this r ? ANS: Substitute for r from L into . Solve to find t and substitute back into L.

18. The Intersection of a Line and a Plane I’ll use column vectors: Substituting in L : The coordinates are Subs. into  :

19. The Intersection of a Line and a Plane Exercise 1. Find the point of intersection of the line L and the plane  where 2.How can you tell that the following line L and plane  don’t intersect?

20. The Intersection of a Line and a Plane Solution: 1. Subs. in  :

21. The Intersection of a Line and a Plane Subs. in L : Coordinates are

22. The Intersection of a Line and a Plane 2.How can you tell that the following line L and plane  don’t intersect? Solution: and the direction vector of the line... If we form the scalar product of the normal vector of the plane... the result is zero showing they are perpendicular. Since ( 1, 2, 1 ) is on the line but not on the plane, the line and plane do not intersect. A x           5 1 2 n The line and plane can only intersect if the line lies on the plane.

23. Finding Angles A x   the direction vector of the line and the normal of the plane. Tip: It’s easy to confuse the procedure for finding angles in the different situations so I always do a sketch. BUT the angle we want is  ( find the acute angle ) The Angle between a Line and a Plane We use the scalar product with

24. Finding Angles These lines...  The angle between 2 planes are perpendicular to the line of intersection of the planes.

25. Finding Angles The quadrilateral has 2 angles adding to  The angle between 2 planes   so,

26. Finding Angles  The angle between 2 planes The angle between 2 planes is equal to the angle between the normal vectors to the planes.  I’ve illustrated the obtuse angle because it’s easier to see. We usually give the acute angle.

27. Finding Angles  The angle between a line L and a plane  is given by SUMMARY where  is the acute angle between the direction vector of the line,, and the normal vector to the plane, n.  The angle between 2 planes  1  and  2  is the angle between their normal vectors and is given by where n 1 and n 2 are the normal vectors to the planes. If necessary subtract from to find the acute angle. Always do a sketch when finding angles. Any line or plane(s) will do.

28. Finding Angles 1. Find the angle, to the nearest degree, between the line and plane given below Exercise 2. Find the angle, to the nearest degree, between the planes given below

29. Finding Angles 1. Solutions: A x  

30. Finding Angles 2. Solutions: Acute angle is  

31. The Vector Equation of a Plane

32. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

33. The Vector Equation of a Plane  The vector equation of a plane is given by SUMMARY where a is the position vector of a fixed point on the plane n is a vector perpendicular to the plane and r is the position vector of any point on the plane. n is called the normal vector or  The Cartesian form is where n 1, n 2 and n 3 are the components of n and

34. The Vector Equation of a Plane e.g.1 Find the equation of the plane through the point A(2, 3,  1) perpendicular to. Solution: Calculating the left-hand scalar product gives the Cartesian form of the equation.

35. The Vector Equation of a Plane A x   The Angle between a Line and a Plane the direction vector of the line and the normal of the plane. Tip: It’s easy to confuse the procedure for finding angles in the different situations so I always do a sketch. BUT the angle we want is  ( find the acute angle ) We use the scalar product with

36. The Vector Equation of a Plane  The angle between 2 planes  I’ve illustrated the obtuse angle because it’s easier to see. We normally give the acute angle. The angle between 2 planes is equal to the angle between the normal vectors to the planes.

37. The Vector Equation of a Plane  The angle between a line L and a plane  is given by SUMMARY where  is the acute angle between the direction vector of the line,, and the normal vector to the plane, n. If necessary subtract from to find acute angle. Always do a sketch when finding angles. Any line or plane(s) will do. where n 1 and n 2 are the normal vectors to the planes.  The angle between 2 planes  1  and  2  is the angle between their normal vectors and is given by