1. Todays Plan C1 Marks – Class Average: 76% New Lesson –Midpoint (6.4) –Division Point (6.4) –Scalar Product (6.5) Homework Questions PRACTICE

3. Midpoint of a Line Segment The coordinate of M are: M B A

4. Properties of Midpoint of a Line Segment 1.MA + MB = 0 2.AM = MB 3.AM = ½AB 4.AB = 2AM M B A

5. Division point of a Line Segment Point P divides the line: –in a 3:2 ratio or 3/2 from A or –in a 2:3 ration or 2/3 from B P B A 2 3

6. Division point of a Line Segment We can also say: P is located at 3/5 from point A or P is located at 2/5 from point B P B A 2 3

7. Division point of a Line Segment The vector equality is: AP = 3/5 AB or BP = 2/5 BA P B A 2 3

8. Example 1 Consider A(1,4) and B(7,1), find P if AP = AB AB = (7-1,1-4) = (6,-3) AP = (x P -x A, y P -y A ) We know that AP = AB so fill in the other info… (x P - 1, y P - 4) = (6,-3)

9. Example 1 Consider A(1,4) and B(7,1), find P if AP = AB (x P - 1, y P - 4) = (6,-3) (x P - 1, y P - 4) = (4,-2) Look at x and y individually… x P – 1 = 4 and y P – 4 = -2 x P = 4 +1 and y P = -2 +4 x P = 5 and y P = 2 So point P is (5,2)

11. SCALAR PRODUCT The scalar product of two vectors u and v is a real number defined by: uv = ||u||×||v||×cosӨ u Ө v

12. SCALAR PRODUCT 1.When vectors u and v are orthogonal, their scalar product is ZERO 2.uu = ||u||×||u||×cos0˚ = ||u|| 2

13. Example 2 Consider u and v form a 30˚ angle and that ||u||=4 and ||v||=3. Find the scalar product. uv = 4 x 3 x cos30 uv = 10.39

14. SCALAR PRODUCT IN THE CARTESIAN PLANE Let u=(a,b) and v=(c,d), we have: uv = ac + bd If we know the angle Ө formed by vectors u and v, we have: cosӨ = uv _ ||u|| ||v||

15. Example 3 Let u=(4,1) and v=(2,3). Find the scalar product and the angle between the vectors uv = 4x2 + 1x3 uv = 11 What are ||u|| and ||v||? ||u||=4 2 +1 2 = 17 ||v|| = 2 2 +3 2 = 13 cosӨ = uv _ ||u|| ||v||

16. Example 3 Let u=(4,1) and v=(2,3). Find the scalar product and the angle between the vectors cosӨ = uv _ ||u|| ||v|| cosӨ = 11 _ = 11 = 0.74 17 x 13 221 So Ө = cos -1 (0.74) Ө = 42.3˚

17. PROPERTIES OF THE SCALAR PRODUCT uv = vu rusv = (rs)uv u(v+w) = uv + uw

18. Last homework questions? (not the quiz questions…)

19. HOMEWORK p. 300 #25, 26 p. 301 #27, 28, 30 p. 303 #1,2,3 p. 304 #6,7,8,9 (how do you know if two vectors are perpendicular? Scalar product is 0!) p.306-307 try them all TAKE HOME QUIZ DUE TOMORROW