# VECTORS

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```					                                                VECTORS
This is an IB unit not found in the Alberta curriculum. The assignment questions are at the end of this
unit as there are no questions in the textbook. As you proceed through the notes, there will be a
statement indicating what questions from the back may now be completed.

A scalar quantity has only a magnitude (a speed of 3 m/s).

A vector quantity has a magnitude as well as a direction.

-   a velocity of 5m/s east.
-   a force of 3 N up.
-   a displacement of 15 km south.
-   an acceleration of 11m/s2 north.

NOTATION:                           B            AB , v , v (boldface)

Vector AB has an initial point at A and a terminal
A                            point at B.

MAGNITUDE of a VECTOR                           | AB | represents the MAGNITUDE of AB .
| v | represents the MAGNITUDE of v.

EQUALITY of VECTORS                             Two vectors are equal if they have the same
magnitude and the same direction.

NEGATIVE of a VECTOR                            The negative of a vector would have the same
(additive inverse)                              magnitude but opposite direction.

AB  BA                 directions are opposite

AB  BA                  magnitudes are still equal
Proof:

AB  BC  AC                                                         C
AB  AC  BC
AB  AC  CB             BC  CB            A              B

Page 1                                 Math 20 IB Vectors

u
u+v
u+v
v                                v              v
v

u                                u
u

u + v is called the resultant of u and v.

u+v=v+u

SUBTRACTION of VECTORS                    AB  CD  AB  DC               i.e. 10  6  10   6

Examples:
1.     B                        C                1.   AB  AD                  ANS:    AC
A                      D                     2.   AB  BC  CD             ANS:    AD
3.   AB  DH                  ANS:    AF
F              G                4.   AB  FC                  ANS:    EC
E                   H                    5.   CG  HG  GE             ANS:    DE
6.   DB  BF                  ANS:    HB
7.   AB  AD  EA             ANS:    EC

2. Express in terms of AB, BC and / or CG . (answers are not unique)

a.       AC                                   ANS: AB  BC or        AE  EH  HG  GC

b. AH                                         ANS: BC  CG

c. DB                                         ANS: AB  BC

d. BH                                         ANS: BC  CG  AB

e.       AC  DB                              ANS: AB  AB

f.       CA                                   ANS:  AB  BC

Complete Exercises 1-3

REPRESENTATION OF VECTORS:

1. GEOMETRIC VECTORS                A vector represented by a directed line segment drawn
so that its length represents its magnitude.

2. ALGEBRAIC VECTORS                A vector that is written in rectangular form.

Page 2                               Math 20 IB Vectors
FORMS of a VECTOR

1. POLAR FORM: is of the form (a, b) where a is the magnitude of the vector and b is the
direction from the positive x-axis. (not I.B.)

a 
2. RECTANGULAR FORM: is of the form (a, b) or   where a represents the x-value
 b
and b represents the y-value of the terminal point of the vector. Its initial point is the origin
(0, 0). They are also called algebraic vectors and IB uses them in the column vector
3
form. ie. (3, 4) becomes   .
 4

VECTOR OPERATIONS                                            2-dimensions                    3-dimensions
a  d a  d
a  c  a  c                              
ADDITION                                             b  d  b  d
                        b  e   b  e 
                          c  f  c  f 
               

 a   ka 
 a   ka                            
MULTIPLICATION by a SCALAR Y                              k                             k  b    kb 
10
 b   kb                          c   kc 
   

5

æ÷ æö
2ö 3÷               æ +
2     3ö
÷     æö
5
eg ç ÷+ ç ÷ =
ç ÷ ç ÷              ç
ç       ÷=    ç ÷
ç ÷
ç5÷ ç1 ÷             ç       ÷
÷     ç ÷
ç ÷ ç ø
è ø è ÷              ç
è5 +    ÷
1ø     ç ÷
÷
è6ø                                      X
-10                -5        0             5            10

æö
2      æö
4
eg          ç ÷
2 ç ÷=
ç3÷
ç ÷
ç ÷
ç ÷
è ÷     ç ÷
-5
ç ø÷    è6ø÷

Complete Exercises 4-5
-10

COLLINEAR VECTORSwith a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Created

- vectors that lie on the same line when they are in standard position.
- they will be parallel to each other.
- they either have the same direction or opposite direction.
a    c 
- one will always be a scalar multiple of the other.    k  
b    d

Page 3                                      Math 20 IB Vectors
Eg. Determine k so that the following pairs of vectors are collinear.

a.                                                      b.
5     10              1                            7 5                 30
   k            k                                 ;            k
 7   14               2                            6 k                  7

7m  5
5                 5 30
m                6    
7                 7 7

Complete Exercises 8 & 16

DETERMINING RECTANGULAR FORM of a VECTOR
a                                        d
a                                    c       
If the initial point of a vector is   or  b  and its terminal point is     or   e ,
b    c                               d      f 
                                          
d a
c a            
then its rectangular form is       or  e  b  .
d  b    f c 
       

Consider an initial point of (1,2) on the Cartesian plane. If the terminal point is (4,7), then you moved
 3
3 units on the x-axis and 5 units on the y-axis. This means the vector would be   .
5

Complete Exercises 9
a 
   a b
2   2
VECTOR LENGTH (MAGNITUDE)
b

a 
 
 b   a  b  c (show using rectangular prism)
2   2   2

c 
 

Complete Exercises 6
0
0  
ZERO VECTOR                                     This vector has a magnitude of 0.   or  0 
0  
0

UNIT VECTOR                                     A vector with a magnitude of 1.

Page 4                                    Math 20 IB Vectors
NATURAL BASES for VECTORS                     2-D and 3-D space can be easily built using unit vectors.

1             0
2-D                    i=            j=  
0             1 

1             0            0
                            
3-D                    i = 0         j = 1        k = 0
0             0            1 
                            

*   In V2 and V3 every vector can be written as a linear combination (sum with scalar multiples) of any
2 non-zero, non-collinear vectors (any 3 in V3).
Simplest sets are: i & j in V2
i & j & k in V3

a              1      0 a 0 a 
eg   = ai + bj = a   + b   =        
b              0     1   0   b   b 
Complete Exercises 5

Examples:

1. Find 2 unit vectors that are collinear with:

3                                                                 1 3
a. u    Since u  5 ,                                      ANSWERS ARE   
 4                                                                5  4

1
u  32  42  5       If the length of u is 5, then multiple it byto makes its length =1.
5
The answer can be plus or minus as collinear vectors do not need to
be in the same direction.

 4                                                               5  4 
b. v                                                       ANS:           
2                                                                10  2 

v  (4)2  22  20  2 5
1   5
If the length of v is 2 5 , then multiple it by         to makes its length =1.
2 5 10
The answer can be plus or minus as collinear vectors do not need to
be in the same direction.

2                                                               2 
                                                             14  
c. w   3                                                  ANS:        3
 1                                                         14  
 1 
                                                                 

Complete Exercises 7

Page 5                                  Math 20 IB Vectors
a 
a                                       
INNER (DOT) PRODUCT                                       u=                                        b
b                                      c 
 
d
c                                       
v=                                        e 
d                                      f 
 

On formula sheet                  u v = ac + bd            or u        v = ad + be + cf

u   v = |u| |v| cos            = angle between the two vectors
when in standard position.

u v
On formula sheet                  cos  =                        Be sure to be in Degree Mode!
u v

Proof:
(a, b)
(c  a)2  (d  b)2
a 2  b2
u                      (c, d)
v

c2  d 2

Using law of cosines:

                                                                  2  a  b  c  d  cos
2                  2                    2
(c  a ) 2  ( d  b ) 2           a 2  b2             c2  d 2               2       2       2       2

c 2  2ac  a 2  d 2  2db  b 2  a 2  b 2  c 2  d 2             2  a  b  c  d  cos 
2      2          2       2

2(ac  db)  2           a 2  b2                  
c 2  d 2 cos 

ac  bd  a 2  b 2 c 2  d 2 cos 
u  v  u v cos 

Page 6                                             Math 20 IB Vectors
Examples:

 3     4                                                                                   1
1.   and                    u  v  3  4  3  0  12 OR     18  4  cos 45  = 3 2  4        12
 3    0                                                                                     2

4
The angle between any point on the line y=x [i.e. (3,3)]
and the x-axis is 45 as y=x cuts quadrant 1 down the
middle.

1
cos 45       You will learn this in 30 Pure or 30 I.B.
2

0                                  5

ORTHOGONAL VECTORS are perpendicular to each other.

Since cos 90 = 0, u   v = |u| |v| cos 90 = 0 the inner product ( u  v ) = 0 for orthogonal vectors.

Examples:

1. Find angles between following pairs of vectors.

73
 2  3                                                
a.                                                 b.  0   5 
 7  4                                             3  2
  

uv       2  3  7  4                                uv       7 3  05  3 2
cos  =                                               cos  =       
u v   22  72  32   4                             u v   7  02  32  32  52  22
2                                 2

Since we want the angle, hit 2nd cos 2nd Answer
 22                                           27 
ANS:   cos1               127.18
0
ANS:   cos1          54.89
0

 53 25                                         58 38 

Complete Exercises 10

Page 7                                 Math 20 IB Vectors
2. Find the angle between the given vector and the axis.

7                                                                  7 
a.   and negative x-axis                              ANS:   cos1         156.80
0

 3                                                                58 1 

uv       7  1  3  0
We need a vector on the negative x-axis.           cos  =        
u v   7  (3)2  (1)2  02
2

 1
Choose   . You could choose any point.
0

4
                                                                   4 
b.  5  and positive x-axis                            ANS:   cos1         65.06
0

7                                                                  90 1 
 

uv           4 1  5  0  7  0
We need a vector on the positive x-axis.        cos  =       
u v       42  52  7 2  12  02  02
1
 
Choose  0  . You could choose any point.                   Complete Exercises 11, 12
0
 

3. In ABC, A = (6, 3), B = (9, 9), C = (2, 5), find:

a.   C                                                                                    ANS: 56.31
10
The vector between A and C is (4,-2)                                                               B
The vector between B and C is (7,4)

uv       4  2  7  4       20
cos C =                            
u v   42  (2)2  72  42   20 65                    5        C
cos C = 0.5547
C =56.31 
A
b.   A
0                 5                10
The vector between A and C is (-4,2)
The vector between B and A is (3,6)

uv        4  3  2  6         0
cos A =                                                                             ANS: 90
u v   (4)2  (2)2  32  62   20 45
cos A = 0
A =90                                                      Complete Exercises 13, 14

Page 8                                   Math 20 IB Vectors
4. Geometric figure ABCDE is determined by A(3, 5, 6), B(7, 0, 4), C(13, 4, 6), D(8, 2, 1)
and E(10, 7, 5). Determine all right angles.
* determine inner product  u  v (90 angle if inner product is 0)
4 
 
AB   5                                 AB  BC = 0 so B = 90
 2 
          0  right angle
6 
 
BC   4                                   (Actually it should be BA  BC but if the dot
2 
 
 5       64
 
CD   6                                    product is zero it works either way.)
 5 
 
0  right angle
 2
 
DE   5                              ( AB  BC relates to the supplementary angle to  B)
 4
 
 7       –70
      
EA   12 
 1 
      
4         –90
 
AB   5 
 2 
 

A(5, 4, 2), B(5, 2, 7), C(10, 1, 9), D(3, 5, 4), E(1, 9, 2)   ANS: right angles are <B, <D, <E

Complete Exercises 21, 18 and 19

Page 9                           Math 20 IB Vectors
5. Find k so that:
 3   2                             3  2  1 6  4k  20
   
a.  1   6   20                        12  4k  20                                ANS: 8
4  k                                4k  32                    k 8
   

3m  5
 3     5                                                                                40
b.   and   are collinear                   5            5      40                 ANS:
8     k                              m                  8                              3
3             3        3

3       k                            3k  4  5  2k  1  0
         
c.  4  and  5  are orthogonal          5k  20  0                                  ANS: 4
 2k     1                            5k  20                    k 4
         

(k  2)(k  1)  5  6  0
 k  2  k  1                        k 2  3k  2  30  0
d.               are orthogonal                                                      ANS: 7, 4
5       6                           k 2  3k  28  0
(k  7)(k  4)  0         k  7, 4

Complete Exercises 15 – 17 and 20

REPRESENTATION OF A LINE IN A PLANE BY A VECTOR

r = p + td                                         p represents the position of any point on the line
 x 
d is the reciprocal of the slope, d   
 y 
where p and d are vectors and t is a scalar.

2                                                      0 5
Eg. The Cartesian line y =     x  7 is most easily represented as a vector as r =    t  
5                                                      7  2
 5  5                 5   5 
but can also be represented as a vector as r =    t   or r =    t   etc.
 9  2                 5   2

 5
Note how d    never changes.
 2
 0   5   5                      2
  ,   ,   are all on the line y  x  7
 7 9  5                          5

Page 10                                     Math 20 IB Vectors
VECTOR EQUATION OF A LINE GIVEN 2 POINTS ON THE LINE

Given points A and B where OA = a and OB = b and O is the origin.
OA + AB = OB
so AB = OB – OA = b – a

eg What is the position vector of a line on (–3,7) and (4,9)

 3           4
OA = a =   , OB = b =   ,
7            9
 4   3   7 
AB = OB – OA = b – a =        
 9   7   2

eg What is the vector equation of a line on (–3,7) and (4,9)

 3   7                4  7
vector equation is r =    t   or        r =  t 
 7   2                 9   2

 3   2 
1.   The line r =    t   yields points like (3, 4) at t = 0, (1, 11) at t = 1, and (5, 32) at t = 4.
 4  7 
Determine the Cartesian form of the line.

7
y    xb
2
7                                                                           7    29
4    (3)  b                                  ANS: 7x + 2y  29 = 0 or y         x
2                                                                            2     2
29
b
2

2. Find a vector form for the line 2x  3y + 12 = 0.

2 x  12  3 y
2
x4 y
3

2                  0 3           3 3
ANS: line is y       x4      so r     t   or r     t  
3                  4  2           6  2

Page 11                                Math 20 IB Vectors
17   20 
3. Position in km of a helicopter is given by r =     t    where ‘t’ is the number of hours
 11  21 
0
after 8:00 a.m. Sherwood Park is at   . Find:           velocity vector
0
a. distance from Sherwood Park at 10:00 a.m.

10:00 am is 2 hours after 8:00 am, thus t =2.
17   20   23 
r=        2           
 11  21   31 
 23 
ANS:          1490 = 38.60
 31 
b. time when the plane is 123 km west and 133 km north of Sherwood Park
 123  17   20 
ANS: never, r =             t     
133   11  21 
 140   20 
        t  
144   21 
140
t         7 7 x21  144  147
20
This means the plane is NEVER 123 west and 133 north.

 20 
  (20)  21  29 km/h
2    2
c. the speed of the helicopter                         ANS: 
 21 

Questions 30-34 are from recent IB exams and are all excellent.

Can also complete the page before the answers at end of booklet
REPRESENTATION OF A LINE IN THE PLANE BY A VECTOR

Page 12                             Math 20 IB Vectors
EXERCISES (Questions 30-34 added Nov 2010)
B                       C
1. ABCD represents a parallelogram. State a single
vector equal to each of the following.

a. AB  AD                              b. BD  BA
A                           D

c. AC  BA  CB

P                               U
2. A. The diagram on the right represents a rectangular prism.
State a single vector equal to each of the following.                        Q                               V

a. RQ  RS                              b. RQ  QV
T
W
c. RW  RS                              d. RQ  RT                          R                                   S

e.    RQ  RS  VU                              
f. RQ  RS  VU       
g. PW  VP                              h. PT  WU

B. Express the following in terms of QP and/or PV .

a. QV                          b. QU  VP                         c. UP
3. The diagram shows a square-based right pyramid. State                                               A
any vectors equal to each of the following.

a. CB                                   b. AB

c. AB  BC                              d. AB  AE
E
B
e. DE  DA                              f.    AE  ED   DC
C                       D

4. Express the following as a single vector.
2        3               9  3   2 
 2  4               3      2                                                
a.                 b.    4               c. 5  6   2  0             d.  2    4   3  7 
3 6                  2    7                    3     4                  0   9   1 
                               
2 
 3                    
5. u =   , v = 4i – 5j, w =   3  , x = 4i + j – 7k. Find:
1                      1 
 
a. u + v                b. 3u – 7v                c. w + 3x                         d. 5w – x

Page 13                                  Math 20 IB Vectors
6. Determine the magnitude of the following vectors.

5                          4 
 3                   5                                                   
a.                    b.                       c.  12                   d.  2 
0                     3                         84                      6 
                           

7. Find two unit vectors collinear with each of the following vectors.

 33                      1 
5                      2                                                 
a.                    b.                       c.  56                    d.  3 
 12                   1                       72                        1 
                           

8. Determine which of the following vectors are collinear.

 3          8           9            3
a. u =   , v =       , w =           , x =     , y = 40i + 30j
4             6          12          4

1            2        0           1 
                                  
b. u =  3  , v =   6, w =   3 , x =     3  , y = 2i – 3j + k
2            4         2         2 
                                  

9. Given that P is the initial point and Q is the terminal point of PQ , determine the rectangular form
of PQ .
a. P = (3, 7), Q = (0, 8)                                 b. P = (5, 6), Q = (4, 8)

c. P = (2, 0, 8), Q = (4, 11, 0)                         d. P = (3, 2, 7), Q = (4, 8, 9)

*** Determine all angles to the nearest tenth of a degree.
10. Find the angle between the following pairs of vectors.
4  5                         1   3 
3       5             1    3                                                     
a.   ,               b.   ,                        c.  0  ,  3                d.  1  ,  1 
 3     0            4       1                     5   4                      4  0
                               

Page 14                                  Math 20 IB Vectors
11. Find the angle between the given vector and axis.

5                                                     3 
a.   and the positive x-axis.                        b.   and the negative y-axis.
 2                                                   8 

 2                                                  4 
                                                      
c.  3  and the positive z-axis.                     d.  3  and the negative x-axis.
5                                                    2 
                                                      

12. Find the angle the line segment from P(4, 1, 3) to Q(5, 0, 3) makes with the positive y-axis.

13. Find P in        PQR given P (2, 1), Q(3, 7) and R(3, 1).

14. Find R in        PQR given P(3, 1, 1), Q (1, 7, 4) and R(1, 1, 3).

15. Find k so that:
 1   4                           k     2 
3  5                                                                          
a.     = 0                          b.  k    1  = 2                  c.  3       1  6
 k   4                             2      1                         4     k 
                                          

16. Find k so that the following pairs of vectors are collinear.
8           3.2 
 7   4                                     
a. 3i – 9j; 2i + kj                        b.   ;                          c.  3  ;     k 
 k   3                         2           0.8 
            
17. Determine the values of k so that the following pairs of vectors are orthogonal.
3  k 
 5   2                                                              
a.   ,                     b. 6i – j, ki + 3j                     c.  k  ,  2k 
 7   3k                                                            4   5 
   

d. PQ and RS for P(1, k), Q(4, 3), R(5, 2) and S(0, 3).

e. PQ and RS for P(3, 2, 2k), Q(1, 5, 2), R(0, 3, 1) and S(2, 2, 5)

18. Is     PQR right angled for P(6, 5), Q(1, 3) and R(3, 2)?

19. Is     PQR right-angled for P(1, 3, 2), Q(0, 3, 4) and R(2, 0, 1)?

Page 15                            Math 20 IB Vectors
5                               1              2 
                                               
20. Find h and k so that u =  h  is orthogonal to both v =    3  and w =   1  .
k                               2             4 
                                               

21. Determine all the right angles in geometric figure ABCDE given A(2, 3, 5), B(7, 2, 1),
C(3, 1, 3), D(5, 5, 1) and E(1, 6, 4).

IB EXAM QUESTIONS
7         2 
22. The vectors u and v are given by u =   and v =   . Find:
8          4 
a. |u – v|.

0 
b. constants x and y such that xu + y v =   .
 44 

k         1 
23. Two vectors are given by p =   and q =   , k  R. Find:
1         k

a. the value of k for which p and q are mutually perpendicular.

b. the two values of k for which the angle between p and q is 60.
1          5 
24. The position of points A and B are given by position vectors OA    and OB    .
 3          1
a. Find the distance between A and B.

b. Find the size of  AOB to the nearest tenth of a degree.

25. A line passes through the two points (1, 2) and (3, 7). Another line passes through (1, 2) and
(4, 8). Find the acute angle between the two lines to the nearest tenth of a degree.

6          4t  4 
26. Two vectors a =   and b =          have equal lengths. Find the two possible values of t.
 4        2t  5 

Page 16                                   Math 20 IB Vectors
 3         2 
27. Two vectors are given by a =   and b =   .
 5          3 
a. Write down the square of the length of a.

b. Calculate a b, the scalar product.

0
c. Find the vector c such that a + b – c =   .
0

 2          4 
28. Two vectors are given by p =   and q =   .
 3         5 
a. Find the value of the angle between p and q to the nearest tenth of a degree.

b. Find the value of the constant k  R such that p + kq is parallel to the x-axis.

6            3 
29. Two vectors a =   and b =   and a point P with coordinates (3, 2) are given.
 1         4 
Q and R are points such that PQ = a and RP = b. Find:
a. the coordinates of Q and R.

ˆ
b. the angle QPR in the triangle PQR to the nearest tenth of a degree.

30. Nov 2009 Paper 1 (no calc). 3 marks each section.

2             3
               
a. Let u   3  and w   1 . Given that u is perpendicular to w, find the value of p.
 1            p
               
1
 
b. Let v   q  . Given that v  42 , find the possible value of q.
5
 

Page 17                               Math 20 IB Vectors
31.   Nov 2009 Paper 2 (calculator allowed). 7 marks.

3 1                     3   1 
                          
Let L1   3   s  2  and L2   9   t  2  . Let  be an obtuse angle between
8  3                    2   3 
                          
L1 and L2 . Calculate the size of  .

32.   May 2009 Paper 1 - Timezone 1 (no calc). Part A = 3 marks and Part B = 7 marks.

The vertices of the triangle PQR are defined by the position vectors

4         3        6
                    
OP   3  OQ   1 OR   1
1         2        5
                    

a. Find PQ and PR .

1
b. Show that cos RPQ        .
2

33.   Nov 2008 Paper 1 (no calc). Part A = 4 marks and Part B = 2 marks

A particle is moving with a constant velocity along line L. Its initial position is A(6, –2, 10)
and after one second it has moved to B(9, –6, 15).

a. Find the velocity vector AB and find the speed of the particle.

c. Write down a possible vector equation of the line L.

Page 18                               Math 20 IB Vectors
34.    Nov 2008 Paper 2 (calc). Part A = 5 marks, Part B = 3 marks and Part C = 7 marks.

The diagram shows a parallelogram ABCD.                       C

diagram not to scale
B

D

A
The coordinates of A, B, and D are A(1,2,3) B(6,4,4) and D(2,5,5)

5                                      6
                                        
a. Show that AB   2  , find AD and hence show that AC   5  .
1                                       3
                                        

b. Find the coordinates of point C.

c. Find AB  AD and hence find angle A.

40. The point A has coordinates (13, 11) and a point B has coordinates (x, y).

a. Find in terms of x and y, the vector AB .

Vector AB is perpendicular to 5i + 2j.

b. By calculating a suitable scalar product (or otherwise) find an equation for the line l
through A and B.

c. Find the coordinates of the point on l which is closest to the origin.

Page 19                                  Math 20 IB Vectors
B
41. * (Like #6 p408) The diagram represents the relative positions
of the centres of three towns Agosham (A), Buckersfield (B)
Buckersfield is 26 km from Agosham and 10 km further                      A
north, and Chetterham is 42 km due south of Buckersfield.

a. How far east of Agosham is Buckersfield?

ˆ
b. Find the exact value of cos( ABC ). Hence or otherwise, find
the distance of Chetterham from Agosham.
C

1                                                   0
Let i =   represent a displacement of 1 km due east and j =  
0                                                   1 
represent a displacement of 1 km due north.
c. Write down the vectors

i. AB                                          ii. AC

d. The range of 'Radio Buckersfield', broadcast from B, is such that it can just be heard at
only one point P on the Agosham-Chetterham road, d km from A. Write down an
expression in terms of d for:
i. the vector AP                             ii. the vector PB

e. Find an expression in terms of d for the scalar product AP PB , and hence or otherwise:
i. find the value of d.

ii. find the range of 'Radio Buckersfield'.

f. A new transmitter is to be set up at E, equidistant from all three towns.
ˆ
i. explain why BEC  2BAC . ˆ

ii. Use part (i) to find how far from each town the transmitter must be.

Page 20                               Math 20 IB Vectors
REPRESENTATION OF A LINE IN THE PLANE BY A VECTOR

1.   Determine a vector equation for the following lines.

a. 5x  7y + 11 = 0.

b. the line that passes through (1, 4) and (3, 1).

 2   2                     13  1 
2.   The position of ship A is given by a =    t   and ship B by b =           t   where the
 7  3                       2   4
distance is in kilometres and t is the number of hours after 9:00 A.M. The base is at the origin.
Find:
a. the position of ship A at 1:00 p.m. relative to the base.

b. distance between the 2 ships at 1:00 p.m.

c. the time when they would collide.

d. the speed of ship A.

 5   2 
3.   Determine the equation of the line given by r =    t   in Cartesian form.
 8   9 

4.      Find the size of the acute angle between x + 3y – 9 = 0 and 2x – 5y + 4 = 0, giving your
Answer to the nearest tenth of a degree.

 2  7           1  4                  6 
1. a. r =    t   b. r =    t   2. a.            (6 km west and 19 km north of base).
3 5             4   5                 19 
b. 10 km c. 2:00 p.m. d. 13 km/h 3.                    9x + 2y + 29 = 0 4. 40.2

Page 21                               Math 20 IB Vectors
1. a. AC      b. BC          c. AA (zero vector)     2. A. a. RV b. RV c. RT                          d. RU
e. RU     f. RU          g. PS h. 2PU B. a. QP  PV b. 2QP c. QP  PV
3. a. DE        b. none      c. AC d. EB e. AE f. AC
 16     12                                               14      6 
6          11                                1                     37                      
4. a.         b.           c.  30  d.  27  5. a.                   b.          c.  0  d.  16 
9           26          23      12           4                   38          22    2 
                                                                        
 33          1 
1 5                 5  2         1            11  
6. a. 3 b. 34 c. 85           d. 2 14    7. a.                 b.              c.   56  d           3
13  12             5  1         97           11  
 1 
 72            
 2       7 
 3       9                   
8. a. u & w, v & y b. v & x             9. a.       b.  c.  11  d.  10 
1         2   8       2 
               
10. a. 45 b. 122.5 c. 90 d. 98.6 11. a. 21.8 b. 159.4 c. 35.8 d. 138.0
15                3                    21
12. 80.7 13. 75.3 14. 144.2 15. a.                 b. 8 c.          16. a. 6 b.             c. –1.2
4                2                      4
10         1            5              11
17. a.        b.        c. 4,     d. 22 e.           18. yes ( Q ) 19. no 20. h = 4, k = 3.5
21         2            2              12
21. A & C 22. a. 13 b. x = 2, y = 7 23. a. k = 0 b. 2  3                  24. a. 4 2 b. 82.9
1 11                                    1                               3
25. 4.8        26. ,                  27. a. 34 b. 9 c.                 28. a. 107.7 b.
2 10                                     8                               5
29. a. Q = (3, 1), R = (0, 2) b. 43.7 30. a. p = 3 b. q  4         31. 106.6
 1         2
                            PQ  PR 1
32. a. PQ   2  PR   2  b. cos P                =
1           4               PQ PR 2
            
 3                                     6  3                     9  3
                                                                  
33. a. AB   4  , speed = 50  5 2 b. L  r   2   t  4  or L  r   6   t  4 
 5                                     10   5                   15   5 
                                                                  
1                     6
                       
34. a. AB = B – A, AD  D  A   3  , AC  AB  AD   5 
 2                     3
                       
7
b. OC  OA  AC   7  c. AB  AD  13 , cos A                      A  50.6
 
 x  13 
40. a.                 b. 5x + 2y – 87 = 0    c. (15, 6)
 y  11 
5                      24        24             0.6d            24  0.6d 
41. a. 24 km        b.      , 40 km      c. i.   ii.            d. i.            ii.                
13                     10         32            0.8d          10  0.8d 
e. i. 6.4       ii. 25.2     f. i. central angle is equal to twice the inscribed angle      ii. 21.7 km

Page 22                                   Math 20 IB Vectors

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