Determine graphically the sum of two or more vectors.

Establish a coordinate system in problems involving vector quantities.

Use the process of resolution of vectors to find out the components of vectors

Determine algebraically the sum of two or more vectors by adding the components of the vectors.

Chapter 4 Vectors (4.1)

Vectors have both magnitude and direction, magnitude is always positive, direction can be + or -

Two types:◦ Graphical…drawing arrows

Always draw tail to tip, utilize n,s,e,w coordinates.◦ Algebraic…d = 50 km southwest, or 10 m east

The resultant◦ A resultant vector is a vector that is equal to the sum of

two or more vectors.◦ Always draw resultant from tail of first arrow to tip of last.

Representing Vector Quantities

+

=

Or

=

Resultant

+A

AB

B

Graphically, the magnitude of the resultant can be found with the Pythagorean theorem

R2 = A2 + B2

Finding the magnitude of a resultant

?6

10

Many situations involve two velocities, for example, if you are flying in a plane traveling east at 400 m/s and while on the plane you walk to the west at 2 m/s, what is your speed relative to the plane? Relative to the ground?

Likewise if you are aboard a bus traveling 15 m/s to the north and you are walking north at 2 m/s while on the bus what is your speed relative to the bus? Relative to the ground?

Relative velocities: (draw these situations in

notes)

Many times these problems involve boats on a river or plane with a cross wind. Whenever you analyze such a problem draw two vectors, one for the velocity of water or wind, the other for the velocity of the boat/plane.

Relative velocities:

V plane relative to air

V plane relative to ground

V air relative to ground

Vector’s can be broken down into components. For example vector A can be broken into a component that lies in the x direction and one that lies in the y direction giving you components: Ax and Ay

Vector Components (4.2)

Ax

Ay

The components lie on x and y axis where:

+ x is east + y is north - x is west - y is south

Direction (Ɵ) is typically assigned in degrees, going

counterclockwise from east.

Ɵ

Finding magnitude of a vector from components

Depends on what side you are given and what you are trying to find.

40

side Ax

side Ay

A

40°

If given the two components Ax and Ay use Pythagorean theorem

If given the hypotenuse (resultant), the direction in degrees and trying to find one of the components use one of the trig functions Sin or Cos…the formulas are

Ax = hyp cos Ɵ Ay = hyp sin Ɵ

Using Trig to find sides

Practice

15 N

8 N

30°

A 10 kg box is being pulled by a rope with a 15 N force at 30° relative to the horizontal. What is the magnitude of force pulling the box to the east?

Two or more vectors may be added by first resolving each into its x and y components.

The x components are added Rx = ax + Bx + Cx + …

The y components are added Ry = Ay + By + Cy + …

Algebraic addition of vectors

The direction (Ɵ) of a resultant can be found using the formula

Ɵ = sin-1 (Ay/hyp) Ɵ = cos-1 (Ax/hyp) Ɵ = tan-1 (Ay/Ax)

Finding direction of a resultant

An airplane is flying at 250 km/hr towards the south. It encounters a crosswind from the west with a magnitude of 15 km/hr. What is the planes resulting velocity?

Practice