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- 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,)

Trigonometry (Pythagorean Theorem) c2 = a2 + b2

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.

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.

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

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 = 9 + 16 = 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

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)

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.

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

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.

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

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 + 2.0 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 = 16 + 4 = 20  R = 4.5m tanθ = (Ry/Rx) = (– 2/4) = – (1/2) = – 0.5  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 = (15 + 23 – 13)i + (30 – 4 + 15)j + (12 – 5 + 0)k R = 25i + 31j +7k R = dxi + dyj + dzk R2 = dx2 + dy2 + dz2 = (625) + (961) + (49) = 1635  R = 40 cm Example 3.4: The Resultant Displacement (Page 68): Solution

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 + 34.6j……………..………….(2) Resultant = R = A + B = (17.7i – 17.7j) + (20i + 34.6j) = (17.7 + 20)i + (34.6 -17.7)j = 37.7i + 16.9j = Rxi + Ryj Magnitude of R = R2 = Rx2 + Ry2 = (37.7)2 + (16.9)2 = 1706  R = 41.3 km cosθ = (Rx/R) = (37.7/41.3) = 0.9  θ = 24.1o (northeast)