الثانية المحاضرةوالثالثةVectors

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)

Problems: (4,6)

Exercises

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.)

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A vector quantity ( or A ) is completely specified by a number and appropriate units plus a

direction.

Examples: )Displacement , velocity, acceleration and area)

A

The magnitude of the vector A is:

1 -written either A or

2 -has physical units.

3 -is always a positive number.

AA

A

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.

There are three Methods to Adding vectors:

1 -Graphical methods

2 -Algebraically Method .

3 -Vector Analysis

•Adding Vectors

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

BR=A+B

A

B

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.

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

B

(A+B

)

(A

+B+)

C

2 -A+(B+C)

B

(B

+ C

)

A+)

B+C

(

Conclusion (2):

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.

Keep in mind that:

Example :

A is a velocity vectorB is a displacement vector

Find : A+B

Answer:A+B has no physical meaning

A

B

Negative of a Vector

A + (-A) = 0.

The vectors A and -A have the same magnitude but point in opposite directions.

A

A -

Subtracting Vectors

A - B = A + (-B)

A

B

-B

A-

BA - B

Or

A

B

A-B

Find (A-B)

Quick Quiz

)a (14.4 units, 4 units

)b (12 units, 8 units

)c (20 units, 4 units

)d (none of these answers.

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?

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 .

Problems:

)9,12,14,15,16(

Exercises

2 -Algebraically Method.

- The magnitude of R can be obtained from:

- The direction of R measured can be obtained from the law of sines:

: is an angle between A and B

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

60A 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

ˆ| i | = | ˆj | = | ˆk | = 1.

The unit–vector notation for the

vector A is:

A = Axˆi + Ayˆj

Unit Vectors

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:

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.

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 (2)

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 (5)

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.

QUESTION (8)

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.

QUESTION (9)

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(