a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a

Transcription

1 Bellwork a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1

2 Day 1 I. Vector Vocabulary A.) Scalar- This has magnitude but no direction. B.) Vectors- These have both magnitude and direction. Examples Forces, accelertion, momentum, weight and translations in geometry May 15 12:20 PM 2

3 C.) Equality 2 or more vectors are equal if they have the same magnitude and direction. D.) Zero Vector 0 = E.) Addition Let a = and b = then a + b = F.) Subtraction Let a = and b = then a - b = May 18 10:03 AM 3

4 G.) Scalar Multiplication Let a = and k is a scalar then ka = Examples 1.) Let a = b = Find a.) 5 a b.) -b + 3a Apr 20 1:30 PM 4

5 II.) Position vectors Position Vector- a vector with the additional property that it is fixed at its back end to the origin. B( b 1, b 2 ) A ( a1, a 2) AB = AO + OB = b - a So AB = May 18 10:03 AM 5

6 Example 1.) Given P( 2,5) and Q(3,-1) a.) write P and Q as a position vectors 2.) Given AB = and BC = find AC 3.) What if BA = and BC = find AC May 18 10:10 AM 6

7 III. Magnitude A.) Magnitude is the length or the magnitude of a vector is the absolute value. Formula **only 3 dimensional given on IB a = v = = remind you of distance formula? May 15 12:21 PM 7

8 Examples 1.) Find the length of the following vectors a.) b.) May 19 3:05 PM 8

9 try... 2.) Given A= ( 3,-2) and B=( 1,2) a.) OA b.) AB c.) AB Apr 20 1:52 PM 9

10 Bellwork Let a = 2 and b = Find a.) a b.) b c.) a + b d.) a + b e.) Show that a + b < a + b May 21 1:32 PM 10

11 Day 2 III. 3 Dimensional A.) Magnitude is the length or the magnitude of a vector is the absolute value. a = v = = yesterday a = v = Formula **only 3 dimensional given on IB May 15 12:21 PM 11

12 B.) Distance and midpoint 1.) recall 2 dimensional 2.) 3 dimensional AB = [AB] = ( x 2 + x 1, y 2 + y 1, z 1 + z 2 ) So USE FORMULA sheet Apr 26 9:48 AM 12

13 Examples 1.) Find the length of the following vectors a.) Think about what we did yesterday b.) If they don't ask in words, then they use May 19 3:05 PM 13

14 2.) Find the position vectors given A= ( 3,-2,5) and B=( 1,0,-2) a.) OA b.) AB c.) BA You try... d.) [AB] 3.) Given A= ( 2,2,4) and B =( -1,1,2) find a.) AB b.) AB c.) [AB] Apr 20 1:52 PM 14

15 Bellwork P= ( -1,2,4) and Q=( 0,2,5) a.) What are P and Q? Points or Vectors b.) Find PQ c.) Find PQ d.) Find [PQ] Apr 25 12:21 PM 15

16 Geometry Review Slope Formula Parallel lines y= 2x + 3 2x + y = 5 2x - y = 6 Perpendicular lines y = 3x - 1 3y = x + 2 3y = 4 - x Apr 4 7:22 AM 16

17 Day 3 IV. Parallel vectors A.) Vectors a and b are said to be parallel if there is a scalar number k such that a = kb. Examples 1.) Show that a = and b = are parallel. May 19 2:55 PM 17

18 2.) Find the values of a and b where the vectors below are parallel a b ) Let a = 2 and b = 3 1 Find the scalar factor k where a = k b. May 19 2:57 PM 18

19 Bellwork Let a = and b = Find x and y where a and b are parallel Apr 26 12:19 PM 19

20 Day 4 II. Unit Vectors A.) A Unit vector is a vector with a length of 1. ex: 2D i = 1 0 j= 0 1 recall distance formula or magnitude 3D i = j = k = component form B.) A zero vector is a vector quantity with no direction. ex 0 0 May 15 12:20 PM 20

21 C.) Finding Unit Vectors 1.) v = v 1 v 2 = v 1 i + v 2 j unit vector form a i + b j = ( ai + bj) = 1 v v = v v This vector will have length 1. Examples Find a unit vector in the same direction as 1.) 5 i - 2 j 2.) ) Find the value of x when x is a unit vector ¼ May 15 12:21 PM 21

22 4.) Finding a unit vector, b, in the same direction Formula k a a Example Find a unit vector, b, if the length is 5 in the same direction as Apr 26 10:07 AM 22

23 C.) Parallel unit vectors formula = ± k a a 1.) Find the unit vector that are parallel to a.) b.) c.) 6i - 5j May 15 12:22 PM 23

24 Bellwork Given the following vector find, in simplest radical form : a.) a unit vector b.) a unit vector traveling in the opposite direction c.) a vector traveling in the same direction as the given but having length of 6 May 13 3:23 PM 24

25 Bellwork find the length: u = 2i - 6j +2k May 28 8:43 AM 25

26 Day 6 VIII Products A.) Inner Product, scalar, or dot product Let v = and w = v w = v 1 w 1 + v 2 w 2 true also for 3 dimensions Let v = v 1 v 2 v 3 and w = w 1 w 2 w 3 * given on IB exam v w = v 1 w 1 + v 2 w 2 May 15 12:22 PM 26

27 Examples Find the inner product of a and b where 1.) a = ( 3,5) and b = ( 8,-3) 2.) a = ( 2,-1,3) and b = ( 5,3,0) May 21 1:37 PM 27

28 B.) Angles v w = v w cos θ This is used to find angles between vectors. rework the formula v w = v 1 w 1 + v 2 w 2 so v 1 w 1 + v 2 w 2 v w = cos θ or simply May 15 12:22 PM 28

29 Find the angle between the vectors v= -i + 3j and w = -i + 2j First Find the inner product Second Find the magnitude of each Third Plug into formula v 1 w 1 + v 2 w 2 v w = cos θ May 21 1:42 PM 29

30 v 1 w 1 + v 2 w 2 + v 3 w 3 v w = cos θ example # May 21 1:44 PM 30

31 bellwork What is the angle between the vectors and Apr 4 11:55 AM 31

32 Day 7 C.) Perpendicular Vectors If the inner product is equal to zero, then the vectors will be perpendicular. Why? graph y = cos x Examples: 1.) Determine if a and b are perpendicular vectors a = b= 2.) Determine if a and b are perpendicular vectors a= b= May 15 12:23 PM 32

33 3.) How about and? 4.) Find the value of x which will make the vectors perpendicular? 5.) Given the vertices of triangle KLM K( 4,-2,7), L( 6,1,-1), and M ( 3,2,-2), find a.) the position vectors KL and LM b.)measure of < KLM Jun 1 9:57 AM 33

34 6.) Given the vertices of ABC using scalar products, determine if it is a right triangle. A(-2,6) B(4,5) C(1,-4) May 16 8:57 AM 34

35 Bellwork Find the values of x which will make the 2 vectors perpendicular Apr 25 12:23 PM 35

36 Day 8 D.) Equations of lines slope = gradient direction vector = notice relationship with slope Examples 1.) Given two lines 2x - y = 6 and x + 3y = 4. Find the angle between these 2 lines. Apr 26 1:48 PM 36

37 2.) same question use this time x - y= 3 and 3x + 2y = 11 Apr 26 1:51 PM 37

38 Bellwork Two vectors are defined as a = 2i + xj and b = i - 4j Find the value of x if 1.) the vectors were parallel 2.) the vectors were perpendicular Jun 1 9:59 AM 38

39 bellwork p = q = 1.) Find p + q 2.) p - 1/2 q 3.) 4.) p + q May 21 9:30 AM 39

40 Bellwork Given: and Find the exact value of: May 15 3:00 PM 40

41 Bellwork Given the vertices of ABC using scalar products, determine if it is a right triangle. A(-2,6) B(4,5) C(1,-4) May 16 8:57 AM 41