Unit 1: VECTOR CONCEPTS
Introduction
Scalar Quantity-quantities with magnitude only (example is temperature)
Vector Quantity-a quantity having both magnitude and direction (example is wind velocity)
Representations of Vectors

u


u represents the magnitude of the vector u
A point to point vector
A
Vector AB
B
magnitude AB
Equality of Vectors
Two vectors are equal if and only if their magnitudes and directions are the same.
A
C
Opposite of a Vector
Two vectors are opposites if they have the same magnitude but point in opposite directions.
We write one as the negative of the other (for example AB  CD )
B
D
Parallel Vectors
Two vectors are parallel if their directions are either the same or opposite.
The Angle between two Vectors
The angle between two vectors is the angle 0    180 formed when two vectors are placed tail to tail.

A Unit Vector

A unit vector is a vector with magnitude one. A unit vector in the direction of any vector v can be found by


dividing v by its magnitude v :

v
vˆ   Read v hat - represents a unit vector.
v

v 3

v
vˆ  1
v̂
N-E-S-W Directions
Example N 30W
N
W
Bearings
Measured clockwise from north
Example N 30W is equivalent to a bearing of 330
E
S
Geometric Vectors
Vectors that are given a magnitude and a direction using N-E-S-W notation or bearings.
Example


Vector w
w  10 at S 40E
Algebraic Vectors


If we let w be a vector in the plane, we can move w until its initial point is at the origin. Then its endpoint will fall

on some point P with coordinates a, b  . We write w  OP  a, b  . This is called an algebraic vector.
Page 1 of 11.
Homework on vocabulary and introduction
3. An airplane’s direction is E 25 N. Explain
why this is the same as N 65 E or a bearing of 65 .
11. A cube is constructed from three vectors a,b, and c
as shown below.
a) express each of the diagonals AG, BH ,CE and DF
in terms of a,b, and c
b) is
AG  BH ? Explain.
Page 2 of 11.
Answers:
Page 3 of 11.
Geometric and Algebraic vectors
Geometric Vectors
Vectors are given a magnitude and a direction using N-E-S-W notation or bearings.
Example



Vector w
w  10 at S 40E  w = 10 S 40E
Algebraic Vectors


If we let w be a vector in the plane, we can move w until its initial point is at the origin. Then its

endpoint will fall on some point P with coordinates a, b  . We write w  OP  a, b  . This is called an
algebraic vector.
The Cartesian coordinate system for a plane (two dimensions) is constructed from two real number lines –
the x-axis and the y-axis – placed at right angles in the plane. Any point P in the plane is identified by an
ordered pair of real numbers (a, b), which are its coordinates.
Example: Given the algebraic vector OP   3, 7 ,
a) Sketch the corresponding position vector on the Cartesian Plane
b) Express the vector as a geometric vector.
c) Express the vector in unit vector notation.
Page 4 of 11.
Practice for geometric and algebraic vectors:
Answers:
1. Geometric gives magnitude and direction explicitly using compass bearings or compass rose, algebraic
gives coordinates of head of vector when tail is at origin in coordinate grid (also called component form)
2. -5i +2j, 6j, -1i + 6j (hats on i and j) 3. (2, 1), (-3, 0), (5, -5)
need more answers....
Page 5 of 11.
Homework on operations with vectors:
1. What single vector is equal to each of these sums?
a) PT  TS  SQ
{ PQ }
b) AC  GE  CE
EC
c) EA  CB  DB  AD
d) PT  QT  SR  SQ
AG
PR


2. Find the sum of the vectors u and v if  is the angle between them.






a) u  12, v  21 and   70 {27.5, 24 to v } b) u  3, v  10 and   115 {9.1, 17.38 to v }
3. A tour boat travels 25 km due east and then 15 km S 50E . Represent these displacements in a
vector diagram, then calculate the resultant displacement.
{37.74 km, S14.8E }
4. If â and b̂ are unit vectors that make an angle of 60 with each other, calculate
a) 3aˆ  5bˆ
{4.4}
b) 8aˆ  3bˆ
{9.8}


5. What conditions must be satisfied by the non-zero vectors u and v for the following to be true?
   
   
 
 
a) u  v  u  v
{u  v }
b) u  v  u  v
{ u , v have the same direction}
   
 
 
c) u  v  u  v
{ u , v have opposite directions and u  v }
 
   
d) u  v  u  v
{ u , v have opposite directions}
6. ABCDEF is a regular hexagon with sides of unit length. Find the magnitude and direction of
AB  AC  AD  AE  AF
{6, 60 degrees to AB }


 
 
7. If x  11, y  23 and x  y  30 , find x  y
{20}
8. Find the magnitude and direction of the resultant of two forces of 15 N and 8 N acting at an angle
of 130 degrees to each other. (geometrically and algebraically)
{11.6 N, 32 degrees to 15 N}



 
9. If u  5,  7 and v   2, 3 , find w  6u  4v
{(38, -54)}
10. Show that the three points A(5, -1), B(-3, 4) and C(13, -6) are collinear.
11. Find a single vector equivalent to
a) 2,  4  1, 7 {(3, 3)} b) 51, 4 {(5, 20)} c) 04,  5 {(0, 0)} d)  6, 0  71,  1 {(1, -7)}

 

12. Simplify 2iˆ  3 ˆj  4 iˆ  ˆj .
{ 6iˆ  ˆj }
13. For the set of points, A(2, 0), B(3, 6), C(4, 1), and D(5, -5) determine whether AB is parallel to
CD and whether AB  CD ?
{not parallel but magnitudes are equal}
14. If three vertices of a parallelogram are (-5, 3), (5, 2) and (7, -8), determine all possible coordinates
of the fourth vertex.
{(-3, -7), (-7, 13), (17, -9)}
15. Find x and y if 3x,1  22, y   2,1 .
{ x  2, y  1 }
Page 6 of 11.
Homework on applications of vectors
1. Find the resultant of the following forces:
23 N
{5N Right}
38 N
10 N
2. Find the equilibrant of the following forces:
120 N
{105N Left}
100 N
85 N
3. Consider the situation indicated in the diagram involving a car C travelling at 110 km/h, a bus B
travelling at 95 km/h and a limousine L travelling at 125 km/h.
Car
WEST
EAST
Limousine
Bus
a) Find the velocity of the limousine relative to the bus.
b) Find the velocity of the car relative to the bus.
{220 km/h West}
{15 km/h East}
4. Forces of 15 N and 23 N act at a point at an angle of 130 degrees to each other. Find the
magnitude and direction of the resultant.
{17.62 N, 40.7 to 23N force}




5. The resultant of two forces f 1 and f 2 has magnitude 28N. If f 1 and f 2 have the same magnitude


and act at an angle of 65 degrees to each other, what is the magnitude of f 1 and f 2 ?
{16.6N}
6. Determine the magnitude and direction of the equilibrant of forces of 18N and 22N acting at an
angle of 35  to each other.
{38N at an angle of 160  to the 18N force in a direction away from the 22N force}
7. A 45 kg chandelier is being suspended from 2 ropes of length 5 m and 8 m which are attached to
two points on the ceiling 10 m apart. Find the tensions in the wires.
{271.7 N, 386.7 N}
8. An object of mass 5kg is suspended from a horizontal ceiling by two strings making angles of 35
degrees and 62 degrees with the ceiling. Calculate the tensions in these strings.
{35 degree string has a tension of 23N and the 62 degree string has a tension of 40N}
9. A pilot heads her plane at NW with an airspeed of 500 km/h. If the actual groundspeed of the
plane is 480 km/h at a track of N 35W . Determine the windspeed and the wind direction?
{87.7 km/h from S 63.1W }
Page 7 of 11.
10. A pilot sets her plane at a heading of N 24W with a airspeed of 625 km/h. If the wind is from
N 84E at 80 km/h, determine the groundspeed and the track of the plane?
{605.08 km/h at N16.77W }
11. An ocean cruiser is travelling at 20 knots in the direction S 80W . A sailing boat 8 nautical miles
due west of the ocean cruiser is travelling at 7 knots at S 40E . Find the velocity of the ocean
cruiser relative to the sailboat? Does the ocean liner pass in front of or behind the sailboat?
{24.3 knots at W 4.4N , OC passes behind SB}
12. Find the resultant of the following forces:
65 N at S 35E
50 N at E15 N
110 N at N 55W
35 N due West
13. Diagram these forces and find their resultant:
25 N acting S30  W
17 N acting N70  E
13 N acting N25  W
30 N acting E
11 N acting SE
{45.62 N at N 60.05W }
{37.7 N acting S72  E}
 




14. Given : x  9 and y  4 as illustrated, determine 3x  2 y given that x  2 y  15 .

y
{23.5}

x
Page 8 of 11.
Homework on dot product and work:


 
Given vectors u and v as shown, find: u  v
 

a) v  u
u


b) Pr oj (u onto v ) (in terms of v̂ )
 
 
c) 2u  v   3u  3v 
32

v



y  2iˆ  2 ˆj , x   1,4, z  3,2
Given
A(1,3), B(3,2)
 
Find: a) y  x


b) the angle between x and z
 
 
c) 2 x  y   5x  y 

d) Pr oj x
{144.2}
{ 8.48v̂ }
{165.5}


u  10, v  17
{6}
{137.7 degrees}
{144}
 12 6
(
, )
5 5
AB

Find the work done if a force of 45 N in the direction of r  4,7 moves an object from G(1,6) to
H(7,3).
{251.1 Joules}
Given W(-1,5), X(6,-2) and Y(-2,-7), find:
a) the co-ordinates of the point Z if WXYZ is a parallelogram.
b) the angle in the parallelogram at X.




Given u  2,1 and v  1,3 , find the Pr oj v onto u  as an algebraic vector.

6. Find a vector perpendicular to : a) u   2, 5

b) w   7,5
{Z(-9, 0)}
77{46.3 degrees}
 2 1
, }
{
 5 5
{many answers}

Find the work done by a 20 N force acting in the direction of v  (1,3) , which moves an object from
P(3, 1) to Q(4, 5).
{69.6 Joules}




Given u  5,4 and v   3,1, find the Pr oju onto v  as an algebraic vector.
 57  19 
{ ,
}
 10 10 
Calculate the work done if a force of 100N at an angle of 40 degrees with the horizontal moves an object
horizontally 40 m and then up a ramp 10 m long inclined at 15 degrees to the horizontal. {3154.6 J}
Calculate the work done* if a box is pulled from point A to point B (as illustrated) by a force of 80 N
applied at an angle of 30 degrees to the horizontal. (*find work done on box)
{1230 Joules}
B
80 N
3m
A
16 m
Page 9 of 11.
Review Unit 1
1. Define the angle between two vectors.
2. Given:
A
 
Find a) v  u
B

v
 
b) u  v

u
C
D
3. Find TG  AQ  TP  GP  TQ
4. Given:

u
  
draw  2v  u  w

v

w
5. Find the resultant of the following forces:
30 N
42 N
14 N
6. Find the equilibrant of the following forces:
140 N
125 N
105 N
7. Consider the situation indicated in the diagram involving a car C travelling at 110 km/h, a bus B
travelling at 95 km/h and a limousine L travelling at 125 km/h.
Car
WEST
EAST
Limousine
Bus
a) Find the velocity of the limousine relative to the bus.
b) Find the velocity of the bus relative to the car.
8. If the equilibrant of two forces is 27 N acting at S 40W , what is the resultant?
9. For a force of 150 N at a bearing of 300 , find the vertical and the horizontal component.
10. Forces of 45 N and 35 N act at a point at an angle of 120 degrees to each other. Find the magnitude
and direction of the resultant.
Page 10 of 11.
11. A 45 kg chandelier is being suspended from 2 ropes of length 12 m and 8 m which are attached to two
points on the ceiling 14 m apart. Find the tensions in the wires.
12. The wind is blowing from S80E at a wind speed of 85 km/h. A plane’s actual groundspeed is 480
km/h at a track of S16W . Determine the airspeed and heading the pilot should take?
13. A ship is travelling at 22 knots in the direction N15W . A submarine 12 nautical miles due north of
the ship is travelling at 13 knots at S 35W . Find the velocity of the ship relative to the submarine? Does
the ship pass in front of or behind the submarine?
14. Find the resultant of the following forces:
165 N at S 55E
130 N at E 25N
110 N at N 55W
95 N due West

 

 
15. Given h  7, g  13, 2h  g  18 , determine 3 g  h .


16. Given vectors u and v as shown, find:
 
a) u  v



b) Pr oj (u onto v ) (in terms of v )
c) Pr oj (v onto u) (in terms of û )
 
 
d) u  3v   2u  v 

u
40

v


u  12, v  17



17. Given x  2iˆ  ˆj , y   1,2, z  3,2 , A(-1,3) and B( - 3, 2)
Find:
 

a) y  z
b) BA
c) BA
d) ŷ
e) the angle between x and
 
 



f) 2 x  3 y  z
g) 2 x  y   3x  2 y 

z

18. Draw the position vector u  2,3 . Be sure to draw the x and y, axis and label them.




19. Given u  3,1 and v   2,4 , find the Pr oj u onto v  as an algebraic vector.
Answers:
2.a) BC b) AD 3. TA 5. 2N left 6. 120 N left 7. a) 220km/h West b) 15 km/h west
8. 27 N [ N 40E ]
9.  129.9iˆ  75 ˆj
10. 40.93 at 47.8 degrees to the 45N force
11. 12 m rope 228.9N and 8 m rope 362.8 N
12. 478.63 km/h at S 5.9W
13. 31.94 knots at N 3.2E , ship passes behind the submarine
14. 71.8 N at N 71E

15. 3g  h  38.83
16. a) 156.3 b) 0.54 v c) 13 û d) 202.5
17. a) -7 b) (2, 1) c)
 1 2 
5 d) 
,
 e) 7.1 f) (10, -10) g) 16
 5 5
19. (1, 2)
Page 11 of 11.