Chapter 1 Vectors

Vectors are arrows

What are these vectors?

Magnitude of a vector = Length of the arrow 4 3

What are the magnitudes?

Magnitudes (solution)

Adding and subtracting vectors

Add and subtract

Solution

Notations

Vector Components 5 4 -3

Terminology

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

A quick reminder

Trigonometry

Solution

Write down the following three vectors in i j notation Write down the following three vectors in i j notation. Find the sum of these vectors also. 10o 4.5 5 4 50o 60o

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

Calculating the angles

Why is the shift needed?

(-1) times a vector? 5 4 3

5 4 3 5 4 3

In General

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: Move the arrows until the tail of one arrow is at the tip of the other arrow. Trace out the resultant arrow.

Subtracting Vectors Diagrammatically

Example

Example

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

Adding Vectors 2

Distance & Displacement 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, going East Distance Displacement Distance is actually the magnitude of displacement

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

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

Distance & Displacement

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 5 m/s, going East Speed Velocity

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

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

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

Math: Vector Dot Product (scalar product)

Dot Product Example

Dot Product Example

Vector cross product

Cross product example