1. Types of Coordinate Systems
Chapter 3: Vectors
Types of Coordinate Systems
1- Cartesian coordinate system (x- and y- axes)
x- and y- axes
points are labeled (x, y)

2. 2- Plane polar coordinate system (r and θ)
origin and reference line are noted
point is distance (r) from the origin in the direction of angle ()
points are labeled (r,)

3. Trigonometry (Pythagorean Theorem)
c2 = a2 + b2

4. Example (3.1): Polar Coordinates (Page 60)
The Cartesian coordinates of a point in the (xy) plane are:
(x, y) = (–3.50, –2.50) m,
Find the polar coordinates (r, θ ) of this point.

5. 3.2 Vector and Scalar Quantities (Page 60)
Scalar quantity: Quantity has a unit value with no direction (mass, length, volume, speed, temperature, time, age,…..). Vector quantity: Quantity has a unit value (number) with a direction (force, displacement, velocity, acceleration, …). The vector (A) is denoted by: The magnitude of the vector (A) is denoted by:
Figure 3.4: A particle moves from (A) to (B) along an arbitrary path (broken line).
its displacement is represented as a vector (solid line) from (A) to (B)
Figure 3.5: Four vectors are equal because they have equal lengths and point in the
same direction.

6. 3.3: Some Properties of Vectors (Page 61)
1. Equality of Two Vectors (A = B)
Two vectors (A) and (B) may be defined to be equal (A = B) if they have the same magnitude and point in the same direction.
2. Adding Vectors:
2.1: graphical method
Figure 3.6: To add vector (B) to vector (A)
Resultant vector (R = A + B) is the vector drawn from the tail of (A) to the tip of (B).
Figure 3.8: Geometric construction for add 4 - vectors.
Resultant vector (R), that completes the polygon

7. Walking 3.0m (east) and 4.0m (north)
2.2: Scalar method (Page 62)
Walking 3.0m (east) and 4.0m (north)
x = 3m
y = 4m
(Resultant)2 = (R)2 = x2 + y2 = (3)2 + (4)2 = = 25
R = 5m
the angle = tanθ = (y/x) = (4/3) = 1.33
θ = tan-1 (1.33) = 53o
Figure 3.7
2.3: Commutative law of addition (Page 62): A + B = B + A
Figure 3.9

8. 2.4: Associative law of addition (Page 62)
A + B + C = (A + B) + C = A + (B + C)
A + B + C = A + (B + C) A + B + C = (A + B) + C
2.5: Negative of a Vector (Page 62)
A + (– A) = A – A = 0
2.6: Vector Subtraction (Figure 3.11, Page 63)
A – B = A + (– B)

9. scalar quantities? [Answer: Scalar (a, d, e) – Vector (b, c)]
Quiz 3.1 (Page 61): Which of the following are vector quantities and which are
scalar quantities? [Answer: Scalar (a, d, e) – Vector (b, c)]
(a) your age (b) acceleration (c) velocity (d) speed (e) mass
Quick Quiz 3.2 (Page 63): The magnitudes of two vectors A and B are A = 12 units and
B = 8 units. Which of the following pairs of numbers represents the largest and smallest
possible values for the magnitude of the resultant vector R = A + B?
(a) 14.4 units, 4 units (b) 12 units, 8 units (c) 20 units, 4 units (d) none of these answers.
Quick Quiz 3.3 (Page 63): If vector B is added to vector A, under what condition does
the resultant vector A + B have magnitude A + B?
(a) A and B are parallel and in the same direction. (b) A and B are parallel and in opposite directions. (c) A and B are perpendicular
Quick Quiz 3.4 (Page 63): If vector B is added to vector A, which two of the following
choices must be true in order for the resultant vector to be equal to zero? (a) A and B
are parallel and in the same direction. (b) A and B are parallel and in opposite directions.
(c) A and B have the same magnitude. (d) A and B are perpendicular.

10. Example 3.2 (Page 64): A Vacation Trip
A car travels 20.0 km due north and then 35.0 km in a direction 60.0° west of north, as shown in Figure 3.12a. Find the magnitude and direction of the car’s resultant displacement

11. Multiplying a Vector by a Scalar (Page 65)
For Example: If vector (A)⅓ is multiplied by a positive scalar quantity (-⅓), then the product (-⅓A) is one-third the length of A and points in the direction opposite A.
3.4 Components of a Vector (Page 65)
The components of a vector (A) are (Ax) and (Ay) in x-y coordinates
the components of A are:
Ax = A cosθ
Ay = A sinθ
The magnitude of the vector (A):
Quick Quiz 3.5 (Page 65): Choose the correct response to make the sentence true:
A component of a vector is:
(a) always, (b) never, or (c) sometimes: larger than the magnitude of the vector.

12. Unit Vectors (i, j, k) (Page 66):
1- Unit vector is a dimensionless vector having a magnitude of (1).
2- Unit vectors are used to specify the direction.
3- Magnitude of the unit vector equals (1), that is,
Unit Vectors (i, j, k) (Page 66):
Quick Quiz 3.6 (Page 67): If at least one component of a vector is a positive number,
the vector cannot.
(a) have any component that is negative (b) be zero (c) have three dimensions.
Quick Quiz 3.7 (Page 67): If A + B = 0, the corresponding components of the two vectors A and B must be:
(a) equal (b) positive (c) negative (d) of opposite sign.
Quick Quiz 3.8 (Page 67): For which of the following vectors is the magnitude of the
vector equal to one of the components of the vector?
(a) A = 2i + 5j (b) B = - 3j (c) C = +5k

13. tanθ = (Ry/Rx) = (– 2/4) = – (1/2) = – 0.5  θ = 333o
Find the sum of two vectors (A) and (B) lying in the xy – plane and given by:
A = (2.0 i j) m and B = (2.0 i – 4.0 j) m
R = A + B = (2i + 2j) + (2i – 4j) = (2 + 2)i + (2 – 4)j = 4i – 2j = Rxi + Ryj
R2 = Rx2 + Ry2 = =  R = 4.5m
tanθ = (Ry/Rx) = (– 2/4) = – (1/2) = –  θ = 333o
Example 3.3: The Sum of Two Vectors (Page 68):
Solution
A particle undergoes three consecutive displacements:
d1 = (15i + 30j + 12k) cm, d2 = (23i – 14j – 5.0k) cm, d3 = (– 13i + 15j) cm.
Find the components of the resultant displacement (R) and its magnitude (R).
R = d1 + d2 + d3 = (15i + 30j + 12k) + (23i – 14j – 5.0k) + (– 13i + 15j)
R = ( – 13)i + (30 – )j + (12 – 5 + 0)k
R = 25i + 31j +7k
R = dxi + dyj + dzk
R2 = dx2 + dy2 + dz2 = (625) + (961) + (49) =  R = 40 cm
Example 3.4: The Resultant Displacement (Page 68):
Solution

14. Example 3.5: Taking a Hike (Page 68):
A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.
(A) Determine the components of the hiker’s displacement for each day.
(B) Determine the components of the hiker’s resultant displacement R for the trip. Find an expression for R in terms of unit vectors
Example 3.5: Taking a Hike (Page 68):
Ax = A cos45 = (25) cos45 = 17.7 km
Ay = – A sin 45 = – (25) sin45 = – 17.7 km [negative sign (–) means in the negative Y direction)
A = Axi + Ayj = 17.7i – 17.7j……………………….(1)
Bx = B cos60 = (40) cos60 = 20 km
By = B sin60 = (40) sin60 = 34.6 km
B = Bxi + Byj = 20i j……………..………….(2)
Resultant = R = A + B = (17.7i – 17.7j) + (20i j)
= ( )i + ( )j
= 37.7i j
= Rxi + Ryj
Magnitude of R = R2 = Rx2 + Ry2 = (37.7)2 + (16.9)2 =  R = 41.3 km
cosθ = (Rx/R) = (37.7/41.3) =  θ = 24.1o (northeast)