Chapter 1 Vectors Vectors are arrows What are

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Chapter 1 Vectors

Chapter 1 Vectors

Vectors are arrows

Vectors are arrows

What are these vectors?

What are these vectors?

Magnitude of a vector = Length of the arrow 3 4

Magnitude of a vector = Length of the arrow 3 4

What are the magnitudes?

What are the magnitudes?

Magnitudes (solution)

Magnitudes (solution)

Adding and subtracting vectors

Adding and subtracting vectors

Add and subtract

Add and subtract

Solution

Solution

Notations

Notations

Vector Components 4 5 -3

Vector Components 4 5 -3

Terminology

Terminology

Decomposing a vector Hint: Once you know one side of a rightangle triangle and

Decomposing a vector Hint: Once you know one side of a rightangle triangle and one other angle, you can find all the lengths using cos, sin or tan.

A quick reminder

A quick reminder

Trigonometry

Trigonometry

Solution

Solution

Write down the following three vectors in i j notation. Find the sum of

Write down the following three vectors in i j notation. Find the sum of these vectors also. 10 o 4. 5 5 4 50 o 60 o

Angle of a vector Find the angles the four vectors make with the positive

Angle of a vector Find the angles the four vectors make with the positive x-axis. y 30° x

Calculating the angles

Calculating the angles

Why is the shift needed?

Why is the shift needed?

(-1) times a vector? 5 3 4

(-1) times a vector? 5 3 4

4 5 3 3 5 4

4 5 3 3 5 4

In General

In General

Adding Vectors Diagrammatically You are allowed to move an arrow around as long as

Adding Vectors Diagrammatically You are allowed to move an arrow around as long as you do not change its direction and length. Method for adding vectors: 1. Move the arrows until the tail of one arrow is at the tip of the other arrow. 2. Trace out the resultant arrow.

Subtracting Vectors Diagrammatically

Subtracting Vectors Diagrammatically

Example

Example

Example

Example

Adding vectors 1 to find the Add the three vectors total displacement.

Adding vectors 1 to find the Add the three vectors total displacement.

Adding Vectors 2

Adding Vectors 2

Distance & Displacement Distance: How far an object has traveled Displacement (is a vector):

Distance & Displacement Distance: How far an object has traveled Displacement (is a vector): How far an object has traveled and in what direction

Distance or Displacement? 5 m Distance 5 m, going East Displacement Distance is actually

Distance or Displacement? 5 m Distance 5 m, going East Displacement Distance is actually the magnitude of displacement

Addition of distance / displacement Distance = 4 m + 3 m = 7

Addition of distance / displacement Distance = 4 m + 3 m = 7 m 5 m Displacement 4 m = 5 m, in the direction of the arrow 3 m

Another example Distance = 2 m + 4 m + 2 m + 4

Another example Distance = 2 m + 4 m + 2 m + 4 m = 12 m Displacement = 0 m

Distance & Displacement

Distance & Displacement

Scalar & Vector Scalars (e. g. distance, speed): Quantities which are fully described by

Scalar & Vector Scalars (e. g. distance, speed): Quantities which are fully described by a magnitude alone. Vectors (e. g. displacement, velocity): Quantities which are fully described by both a magnitude and a direction.

Speed or Velocity? 5 m/s Speed 5 m/s, going East Velocity

Speed or Velocity? 5 m/s Speed 5 m/s, going East Velocity

Speed = | Velocity | Speed can be interpreted as the magnitude of the

Speed = | Velocity | Speed can be interpreted as the magnitude of the velocity vector:

Summary Three ways to represent a vector: 1. By an arrow in a diagram

Summary Three ways to represent a vector: 1. By an arrow in a diagram 2. By i, j components 3. By the magnitude and angle You need to learn all! 5 3 4

Multiplying vectors Two different products: 1. Dot product (gives a scalar) 2. Cross product

Multiplying vectors Two different products: 1. Dot product (gives a scalar) 2. Cross product (gives a vector)

Math: Vector Dot Product (scalar product)

Math: Vector Dot Product (scalar product)

Dot Product Example

Dot Product Example

Dot Product Example

Dot Product Example

Vector cross product

Vector cross product

Cross product example

Cross product example