the Further Mathematics network

Transcription

1 the Further Mathematics network 1

2 the Further Mathematics network Further Pure 3: Teaching Vector Geometry Let Maths take you Further 2

3 Overview Scalar and vector product Basic skill sets related to lines and planes Applications of scalar and vector product The scalar triple product, determinants, equations and matrices 3

4 Scalar Product = (2 3) + (3 2) + (4 1) = Get students to get a feel for the scalar product Work in two dimensions to begin with when is it negative, when is it positive, when is it zero? 4

5 Use of scalar product in lighting Angle between normal vector and light source determines how light the surface should appear 5

6 Use of scalar product in lighting 6

7 Use of scalar product in lighting 7

8 Use of scalar product in lighting 8

9 Use of scalar product in lighting 9

10 Lighting a Plane 10

11 Vector Product 2 3 (3 1) (2 4) = [(2 1) (3 4) ] = (2 2) (3 3) 5 Also a b = a b sinθ And a b is perpendicular to both of a and b in accordance with the right hand rule. 11

12 Vector Product The fact that a b = a b sinθ means that the length of a b is the same as the area of the parallelogram made using a and b as the sides. a b 12

13 Use of vector product in video games, what s behind what? 13

14 Use of vector product 14

15 The scalar and vector product a.b = a b cosθ a a.b / a θ a.b / b b 15

16 Or if b = 1 a.b = a b cosθ a θ a.b b 16

17 The scalar and vector product a x b = a b sinθ a x b / b a θ a x b / a b 17

18 Or if b = 1 a x b = a b sinθ a x b a θ b 18

19 In summary To resolve a parallel and perpendicular to b If b is the unit vector of b then a resolved parallel to b is a.b a resolved perpendicular to b is a x b 19

20 Quick Skills Test Resolve 2 in the direction of Resolve 2 perpendicular to

21 The basics The vector equation of a line Conversion between Cartesian and vector form of line 1 2 r = + λ

22 Examples of short questions to ask students 1 2 Is (3, 2) on the line r = + λ? Are the lines μ s = +, r = + λ 2 6 parallel? Find the line which has the points (1, 4) and (7, 10) on it. 22

23 The basics The vector equation of a line 1 2 r = 2 + λ

24 Introducing Planes - Ideas If the Cartesian equation of a line is given in the form ax + by = d then a b is perpendicular to the line and as d changes the height of the line changes. 24

25 Planes ( 1, 1, 2 ) ( x, y, z ) 25

26 Basics list Vector and Cartesian form of equations and planes and how to convert between them. Finding a line given two points on it Finding a plane given three points on it Finding the intersection of two lines. Finding the intersection of a line and a plane Finding the intersection of two planes Finding the intersection of three lines and geometrical interpretation 26

27 Testing the basics Basics jigsaw Timed tests 27

28 Applications of the scalar and vector product Distance between a point and a line Distance between two skew lines Distance of a point from a plane Scalar Triple Product and applications 28

29 Distance between a point and a line We can see this is just AP resolved perpendicular to the direction of l. l A P 29

30 Distance between skew lines We can see that this is just the vector between any point on l 1 and any point on l 2 resolved in the direction perpendicular to both lines l 1 P A l 2 Q B 30

31 Distance of a point from a plane This is just RP resolved parallel to the normal vector where R is any point on the plane. P normal R 31

32 The scalar triple product Where a, b and c are vectors this means a.(b x c). We know that a.(b x c) = a b x c cosθ where θ is the angle between a and b x c. b x c is the area of the base of the parallelopiped and the modulus of a cos θ is its vertical height. Therefore a.(b x c) is the volume of the parallelopiped formed from a, b and c 32

33 Applications Testing whether two lines meet Testing whether four points are coplanar Proving that the determinant gives the scale factor of volume enlargement for transformations of three dimensional space. 33

34 Collision Detection Asteroids was one of the first video games 34

35 A Using vectors in Collision Detection 2D α C γ β P When P is outside the triangle ABC α + β + γ < 360 A When P is inside the triangle ABC, B C α + β + γ = 360 γ α β P B 35

36 Geogebra Demo 36

37 Video Games Vector Tricks 37

38 Exam Question 38

39 Examiners Comments 39

40 Exam Question 40

41 Examiners Comments 41