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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

- 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

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

R=A+B+C+D

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.

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

A

-A

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 ?

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.

(Examples)

- 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

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:

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:

- 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

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

Answer is d

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