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Chapter 2. Motion in two dimensions. 2.1: An introduction to vectors. 2.1: An introduction to vectors. Vectors : Magnitude and direction Examples for Vectors: force – acceleration- displacement Scalars : Only Magnitude

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Chapter 2

Motion in two dimensions

2.1: An introduction to vectors

2.1: An introduction to vectors

  • Vectors: Magnitude and direction

  • Examples for Vectors: force – acceleration- displacement

  • Scalars: Only Magnitude

  • A scalar quantity has a single value with an appropriate unit and has no direction.

    • Examples for Scalars: mass- speed- work-Distance- Energy-Work-Pressure

Motion of a particle from A to B along an arbitrary path (dotted line). Displacement is a vector

  • Vectors:

  • Represented by arrows (example displacement).

  • Tip points away from the starting point.

  • Length of the arrow represents the magnitude

  • In text: a vector is often represented in bold face (A) or by an arrow over the letter.

  • In text: Magnitude is written as A or

This four vectors are equal because they have the same magnitude and same length

Adding vectors:

Two vectors can be added using these method:

1- tip to tail method.

2- the parallelogram method.

1- tip to tail method.

Graphical method (triangle method):

Draw vector A. Draw vector B starting at the tip of vector A.

The resultant vector R = A + B is drawn from the tail of A to the tip of B.

Adding several vectors together.

Resultant vector


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

Commutative Law of vector addition

2- the parallelogram method.

A + B = B + A

(Parallelogram rule of addition)

Associative Law of vector addition

A+(B+C) = (A+B)+C

The order in which vectors are added together does not matter.

Negative of a vector.

The vectors A and –A have the same magnitude but opposite directions. A + (-A) = 0



Subtracting vectors:

A - B = A + (-B)

Multiplying a vector by a scalar

The product mA is a vector that has the same direction as A and magnitude mA.

The product –mA is a vector that has the opposite direction of A and magnitude mA.

Examples: 5A; -1/3A

  • Given , what is ?

Components of a vector

The x- and y-components of a vector:

The magnitude of a vector:

The angle q between vector and x-axis:

The signs of the components Ax and Ay depend on the angle q and they can be positive or negative.


Unit vectors

  • A unit vector is a dimensionless vector having a magnitude 1.

  • Unit vectors are used to indicate a direction.

  • i, j, k represent unit vectors along the x-, y- and z- direction

  • i, j, k form a right-handed coordinate system

The unit vector notation for the vector A is:

OR in even better shorthand notation:

Adding Vectors by Components 1.

We want to calculate: R = A + B

From diagram: R = (Axi + Ayj) + (Bxi + Byj)

R = (Ax + Bx)i + (Ay + By)j

Rx = Ax+ Bx

Ry = Ay+ By

The components of R:

The magnitude of a R:

The angle q between vector R and x-axis:

Example 1.

  • A force of 800 N is exerted on a bolt A as show in Figure (a). Determine the horizontal and vertical components of the force.

The vector components of F are thus,

and we can write F in the form

Example : 1.

The angle between where and the positive x axis is:

  • 61°

  • 29°

  • 151°

  • 209°

  • 241°

W 1.

Example :

F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S


Ex : 2 – 10 1.

A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?

example 1.

Answer is d