Ashley AbidNicole Bogdan

Vectors

Vectors and Scalars

A vector quantity is a quantity that is fully described by both magnitude and direction.

Scalars are quantities that are fully described by a magnitude (or numerical value) alone.

Examples of Vectors Examples of Scalars

Displacement (distance in a direction)

Distance (m)

Velocity (distance over time) Temperature (C or F)

Acceleration (velocity over time)

Energy (J)

Force (newtons * acceleration) Time (s)

Drawing Vectors

All vectors can be represented as arrows.

Tail Head

Magnitude of Vectors in One Dimension

• Vectors acting in the same direction produce the greatest magnitude force

• Vectors acting in opposite directions produce the

smallest magnitude force

• At 0 degrees, magnitudeis greatestAt 180 degrees, magnitudeis the smallest

What is displacement?Displacement is an object's overall change in position. It takes direction into account.

If a person walks around the perimeter of the diagram, the total distance traveled would be4m + 2 m + 4 m + 2 m = 12m

However, the total displacement is calculated as4 m East + 2 m South + 4 m West + 2 m North = 0 m.East and West cancel one another out, just like North and South.

Vector Fundamentals in Two DimensionsVectors can be added together to form a resultant

vector. The vectors added together are called component

vectors. They are represented with compass directions on the x and y axis.

Resultant --> <-- components

Adding Vectors and Calculating Resultants

The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together.

There are two methods to calculate resultants:

Head to TailTail to Tail (Parallelogram Method)

Head to Tail Method

1. Place the two vectors next to each other so that the head of one vector is touching the tail of the other vector.

2. Draw the resultant vector by connecting the remaining head and tail.

V1 + V2 = RV1

V2

Resultant Vector (R)

Head to Tail Method

Often, vectors must be rearranged for the head to tail method. The angles must remain the same.

The Pythagorean TheoremIf two vectors are perpendicular to each

other, you can solve for their resultant using the

Parallelogram Method

1. Draw the two components 2. Extend parallel lines to each of with their tails touching. the components so that

their lines meet

3. The resultant is the diagonal extended from one corner to the next.

V1

V2

R

Trigonometry Review for Parallelogram Method

The hypotenuse represents the resultant force. The adjacent and opposite represent the components.

Finding Horizontal and Vertical Components to a Vector

The vertical and horizontal components make a triangle and so we can use sine and cosine to calculate a missing component.

The formulas Rx =R cosθ and Ry = R sinθ

are used.

Ver

tica

l C

ompon

ent

R

y

Horizontal Component

Rx

θ

Finding an Equilibrant

• Equilibrium is any situation where the net force acting on an object is zero.

• It is called equilibrium because all the forces acting on the object equal out and cancel each other.

• This third force that would do the cancelling out is called the equilibrant.

• The equilibrant is a vector that is the exact same size as the resultant would be, but the equilibrant points in exactly the opposite direction.

• For this reason, an equilibrant touches the other vectors head-to-tail like any other vector being added.

Drawing Equilibrants

The equilibrant is the exact same size as the resultant would be, but the equilibrant points in exactly the opposite direction

(short animation)http://www.stmary.ws/highschool/physics/home/animations3/forces/resultants_90_degrees.swf

Review Question #1

Review Question #1 Solved

Ry = R sinθRy = 300 N * sin(60)=(2) 260 N

Review Question #2

Review Question #2 Solved(3)

Use the parallelogrammethod to find the missing component.

Review Question #3

Review Question #3 SolvedResultant:(F1^2 + F2^2) = R^2R = 14 N North East

Equilibrant is equal inmagnitude butopposite in direction

Equilibrant = 14 N South West (1)

R = 14 N

Review Question #4

Review Question #4 Solved(1) decreases

At 0 degrees, bothobjects work in the same direction. Their

magnitude is added.

At 90 degrees, the pythagorean theorem is used

6 N ^2 + 8 N ^2 = 10 N ^214 N > 10 N.

6 N + 8 N

14 N

Review Question #5

Question #5 Solved

(1) Law of TrianglesThe sum of any two sides of a triangle

cannot be smaller than the third side1 N + 3 N < 5 N