المحاضرة الثانية والثالثة Vectors

(r is the distance an θ is the angle) 3.1 Coordinate Systems 1- Cartesian coordinates (rectangular coordinates). (x, y) 2- polar coordinate system. (r is the distance an θ is the angle)

we can obtain the Cartesian coordinates by using the equations: positive θ is an angle measured counterclockwise from the positive x axis

Example : Polar Coordinates Note that you must use the signs of x and y to find θ

Problem (1)

Problem (2)

Problem (5)

Exercises Problems: (4,6)

3.2 Vector and Scalar Quantities A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. Examples: (volume, mass, speed, distance, time intervals and Temperature).

A vector quantity ( or A ) is completely specified by a number and appropriate units plus a direction. Examples: (Displacement , velocity, acceleration and area)

3- is always a positive number. The magnitude of the vector A is: 1- written either A or 2- has physical units. 3- is always a positive number.

Quick Quiz: Which of the following are vector quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass

3.3 Some Properties of Vectors • Equality of Two Vectors A = B only if A = B and if A and B point in the same direction along parallel lines.

Example: These four vectors are equal because they have equal lengths and point in the same direction.

2- Algebraically Method. 3- Vector Analysis • Adding Vectors There are three Methods to Adding vectors: 1- Graphical methods 2- Algebraically Method. 3- Vector Analysis Adding vectors = finding the resultant (R)

1- Graphical methods

When vector B is added to vector A, the resultant R is the vector that runs from the tail of A to the tip of B. A R=A+B B B A

Example: If you walked 3. 0 m toward the east and then 4 Example: If you walked 3.0 m toward the east and then 4.0 m toward the north, the resultant displacement is 5.0 m, at an angle of 53° north of east.

R = A + B + C + D is the vector that completes the polygon Geometric construction is Used to add more than two vectors The resultant vector R = A + B + C + D is the vector that completes the polygon

R is the vector drawn from the tail of the first vector to the tip of the last vector.

commutative law of addition: 1- A + B R

2- B+A R

Conclusion (1):

Keep in mind that: A + B = C is very different from A + B = C. * A + B = C is a vector sum. * A + B = C is a simple algebraic addition of numbers

Associative law of addition: 1- (A + B) + C (A+B)+C (A+B) B

2- A+(B+C) A+(B+C) (B + C) B

Conclusion (2):

Keep in mind that: When two or more vectors are added together, all of them must have the same units and all of them must be the same type of quantity.

Example : A is a velocity vector B is a displacement vector Find : A+B Answer: A+B has no physical meaning A B

Negative of a Vector A + (-A) = 0. A - The vectors A and -A have the same magnitude but point in opposite directions. A A + (-A) = 0. A -

Subtracting Vectors A - B = A + (-B) Find (A-B) -B Or B -B A B A-B

Quick Quiz (a) 14.4 units, 4 units (b) 12 units, 8 units 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 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 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.

Problem (8) Answer:

Problem (10) Answer:

Multiplying a Vector by a Scalar - If vector A is multiplied by a positive scalar quantity m, then the product mA is a vector that has the same direction as A and magnitude mA. - If vector A is multiplied by a negative scalar quantity -m, then the product -mA is directed opposite A.

Example: The vector 5A is five times as long as A and points in the same direction as A.

Exercises Problems: (9,12,14,15,16)

2- Algebraically Method.

- The magnitude of R can be obtained from: : is an angle between A and B - The direction of R measured can be obtained from the law of sines:

Example : 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. Find the magnitude and direction of the car’s resultant displacement. Solution 60 A B

Graphical method for finding the resultant displacement vector R = A +B. (in the previous example)

3- Vector Analysis

Components of a Vector and Unit Vectors Note that the signs of the components Ax and Ay depend on the angle θ.

The signs of the components of a vector A depend on the quadrant in which the vector is Located.

The magnitude and direction of A are related to its components through the expressions:

Quick Quiz 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.

Example: The components of B along the x' and y' axes are: Bx' = B cos θ' By' = B sin θ'

Unit Vectors A unit vector (ˆi , ˆj, and ˆk )is a dimensionless vector having a magnitude of exactly 1. that is |ˆi | = | ˆj | = | ˆk | = 1.

Unit Vectors The unit–vector notation for the vector A is: A = Axˆi + Ayˆj

Example Consider a point lying in the xy plane and having Cartesian coordinates (x, y), as in Figure. The point can be specified by the position vector r, which in unit–vector form is given by:

- vector A has components Ax and Ay. find R = A + B if: - vector A has components Ax and Ay. -vector B has components Bx and By :

Because R = Rxˆi + Ryˆj, we see that the components of the resultant vector are

The magnitude of R: the angle it makes with the x axis from its components,

Three-dimensional vectors

Quick Quiz 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 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 For which of the following vectors is the magnitude of the vector equal to one of the components of the vector? (a) A = 2i ˆ + 5ˆj (b) B = -3ˆj (c) C = +5k

Example : The Sum of Two Vectors

Example : The Resultant Displacement A particle undergoes three consecutive displacements: d1 = (15ˆi + 30ˆj + 12ˆ k) cm, d2 = (23ˆi - 14ˆj - 5.0ˆ k) cm and d3 =(-13ˆi + 15ˆ j) cm. Find the components of the resultant displacement and its magnitude.

QUESTION (1) Two vectors have unequal magnitudes. Can their sum be zero? Explain. No. The sum of two vectors can only be zero if they are in opposite directions and have the same magnitude. If you walk 10 meters north and then 6 meters south, you won’t end up where you started.

QUESTION (2) Can the magnitude of a particle’s displacement be greater than the distance traveled? Explain. No, the magnitude of the displacement is always less than or equal to the distance traveled.

QUESTION (5) A vector A lies in the xy plane. For what orientations of A will both of its components be negative? For what orientations will its components have opposite signs? If the direction-angle of A is between 180 degrees and 270 degrees, its components are both negative. If a vector is in the second quadrant or the fourth quadrant, its components have opposite signs.

QUESTION (8) If the component of vector A along the direction of vector B is zero, what can you conclude about the two vectors? Vectors A and B are perpendicular to each other.

Can the magnitude of a vector have a negative value? Explain. QUESTION (9) Can the magnitude of a vector have a negative value? Explain. No, the magnitude of a vector is always positive. A minus sign in a vector only indicates direction, not magnitude.

Problem (18)

Problem (28)

Problem (30)

Problem (31)

Problem (39)

Problem (43)

Problem (49)

Problem (50)

Exercises Questions: (3,4,6,11,12,14) Problems: (21, 32,53)(22,25,47,53)