Chapter 1 Vectors Vectors are arrows What are
- Slides: 45
Chapter 1 Vectors
Vectors are arrows
What are these vectors?
Magnitude of a vector = Length of the arrow 3 4
What are the magnitudes?
Magnitudes (solution)
Adding and subtracting vectors
Add and subtract
Solution
Notations
Vector Components 4 5 -3
Terminology
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
Trigonometry
Solution
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 x-axis. y 30° x
Calculating the angles
Why is the shift needed?
(-1) times a vector? 5 3 4
4 5 3 3 5 4
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: 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
Example
Example
Adding vectors 1 to find the Add the three vectors total displacement.
Adding Vectors 2
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 the magnitude of displacement
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 m = 12 m Displacement = 0 m
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 Speed 5 m/s, going East Velocity
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 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 (gives a vector)
Math: Vector Dot Product (scalar product)
Dot Product Example
Dot Product Example
Vector cross product
Cross product example
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