Understanding Vectors: Magnitudes, Components, and Operations Explained
This chapter delves into the fundamentals of vectors, defining them as arrows that represent quantities with both magnitude and direction. It explains how to calculate magnitudes as the length of the arrow and provides methods for adding and subtracting vectors both algebraically and diagrammatically. Furthermore, it covers the essential terminology of vector components and introduces fundamental trigonometric relationships for determining angles. The distinctions between distance and displacement are clarified, emphasizing the differences between scalars and vectors, and summarizing various ways to represent vectors.
Understanding Vectors: Magnitudes, Components, and Operations Explained
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Presentation Transcript
Chapter 1 Vectors
3 4 Magnitude of a vector= Length of the arrow
4 -3 5 Vector Components
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.
Angles of a vector Find the angles the four vectors make with the positive x-axis. y 30° x
Write down the following three vectors in i j notation. Find the sum of these vectors also. 10o 4.5 5 4 50o 60o
5 3 4 (-1) times a vector?
5 4 3 3 4 5
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.
Adding vectors 1 Add the three vectors to find the total displacement.
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 5 m, going East Distance Displacement Distance is actually the magnitude of displacement
4m Addition of distance / displacement Distance = 4m + 4m = 8m Displacement = 0m
5m 3m 4m Addition of distance / displacement Distance = 4m + 3m = 7m Displacement = 5m, in the direction of the arrow
Another example Distance = 2m + 4m + 2m + 4m = 12m Displacement = 0m
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:
5 3 4 Summary 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!
