Chapter 4 Vectors

The Cardinal Directions

Vectors

• An arrow-tipped line segment used to represent different quantities.

• Length represents magnitude.

• Arrow head represents direction.

Vector Addition in 1 - Dimension

• When vectors point in the same direction we add them just as we would add any two numbers.

Vector Addition in 1 - Dimension

• When vectors point in opposite directions we subtract them just as we would with any two numbers.

Vector Addition in 2-Dimensions

• Vectors in 2-dim are added by placing the tail of one to the head of another.

Remember This?

Addition of Several Vectors

• The order of addition is not important.

• R is called the resultant.

Independence of Vector Quantities

• Perpendicular vectors can be treated independently of each other.

Analytical Method of Vector Addition

• The sum of any two vectors can be determined using trigonometry.

Adding Perpendicular Vectors

Angle θ is =

a) 25 deg

b) 14 deg

c) 35 deg

d) 45 deg

•

Angle θ is =

a) 25 deg

b) 14 deg

c) 35 deg

d) 45 deg

Vector Components

• We can take two vectors and replace them with a single vector that has the same effect. This is vector addition.

• We can start with a single vector and think of it as a resultant of two perpendicular vectors called components.

• This process is called vector resolution.

Example

Example 2

Problem Solving Strategy

• In resolving vectors choose the most convenient axis according to the specifics of the problem.

• Choose the axis that simplifies the solution.• Axis may be up-down, left-right, east-west

or north-south.• Be sure to specify the positive direction for

each.

Adding Vectors at Any Angle

• Vector resolution is the method used.

• Resolve all vectors into x and y components.

• Add all x’s and all y’s together.

• Use xtot and ytot to create a right triangle.

• Use Pythagorean formula to calculate resultant and trig to find angle.

R is = ?

a) 15 N

b) 12 N

c) 20 N

d) 11N

R is = ?

a) 15 N

b) 12 N

c) 20 N

d) 11N

Θ is = ?

a) 53 deg

b) 35 deg

c) 25 deg

d) 45 deg

Θ is = ?

a) 53 deg

b) 35 deg

c) 25 deg

d) 45 deg

Applications of Vectors

Vectors can be used to represent:

- displacement

- velocity

- acceleration

- force

Equilibrium

• When the net force is zero, the object is in equilibrium.

• When the vector sum of the forces is not zero, a force can be applied that will produce equilibrium. This force is called the equilibrant.

• It is equal in magnitude but opposite in direction to the resultant.

3 Forces in Equilibrium:

a) produce a net force.

b) produce a triangle for a vector diagram.

c) are called an equilibrant.

d) produce an acceleration.

3 Forces in Equilibrium:

a) produce a net force.

b) produce a triangle for a vector diagram.

c) are called an equilibrant.

d) produce an acceleration.

Gravitational Force and Inclined Planes

• Gravitational force always points towards center of Earth.

• This is weight.

• Choose one axis parallel to the plane and the other perpendicular to it.

Formulas

• R2 = A2 + B2 – 2AB cos Θ

• Ax = A cos Θ

• AY = A sin Θ

• A = Ax + AY