PHYS 2211 Module 2.2

Coordinate Systems and Components of a Vector

Recommended Reading

2.2 Coordinate Systems and Components of a Vector

Learning Objectives

By the end of this section, you will be able to:

  • Describe vectors in two and three dimensions in terms of their components, using unit vectors along the axes.
  • Distinguish between the vector components of a vector and the scalar components of a vector.
  • Explain how the magnitude of a vector is defined in terms of the components of a vector.
  • Identify the direction angle of a vector in a plane.
  • Explain the connection between polar coordinates and Cartesian coordinates in a plane.

Trig Review

Before we get into the discussion about vector components, let’s first do a quick review of trigonometry. Test yourself with the following practice problems.

Practice!

Practice 2.2.1

From the figure, how do you define the ?
(a)
(b)
(c)
(d)
(e)
Practice 2.2.2

From the figure, how do you define the ?
(a)
(b)
(c)
(d)
(e)
Practice 2.2.3

From the figure, how do you define the ?
(a)
(b)
(c)
(d)
(e)
Practice 2.2.4

What is the length of the side of the triangle adjacent to
θ if θ = 30° and the hypotenuse is 2 units?
(a)
(b)
(c)
(d)
(e)
Practice 2.2.5

What is the length of the side of the triangle opposite to
θ if θ = 30° and the hypotenuse is 2 units?
(a)
(b)
(c)
(d)
(e)
Practice 2.2.6

From the triangle in the figure, what is another way to express c?
(a)
(b)
(c)
(d)
(e)

Vector Components

Practice!

Practice 2.2.7

What is the magnitude of the x-component of vector ?
(a) 14.4 m
(b) 10.8 m
(c) 13.6 m
(d) 18.0 m
(e) 12.0 m
Practice 2.2.8

What is the magnitude of the y-component of vector ?
(a) 14.4 m
(b) 10.8 m
(c) 13.6 m
(d) 18.0 m
(e) 12.0 m

Adding Vectors using Vector Components

Practice!

Practice 2.2.9

When vector is added to vector , the resultant is vector such that . = + . What is the magnitude of the x-component of the resultant vector ?
(a) 26.4 m
(b) 10.8 m
(c) 22.8 m
(d) 6.00 m
(e) 2.40 m
Practice 2.2.10

When vector is added to vector , the resultant is vector such that . = + . What is the magnitude of the y-component of the resultant vector ?
(a) 26.4 m
(b) 10.8 m
(c) 22.8 m
(d) 6.00 m
(e) 2.40 m
Practice 2.2.11

When vector is added to vector , the resultant is vector such that . = + . This resultant vector has components and . How do you determine the magnitude of ?
(a)
(b)
(c)
(d)
(e)
Practice 2.2.12

When vector is added to vector , the resultant is vector such that . = + . This resultant vector has components and . How do you determine the direction of , the angle the vector makes with the positive x-axis?
(a)
(b)
(c)
(d)
(e)
Practice 2.2.13

Vectors and have equal magnitudes but point in different directions, as shown in the figure. What is true about the -component of vector compared to the x-component of vector ?
(a) is equal in magnitude and points in the same direction to .
(b) is greater in magnitude and points in the opposite direction to .
(c) is greater in magnitude and points in the same direction to .
(d) is equal in magnitude and points in the opposite direction to .
Practice 2.2.14

Vectors and have equal magnitudes but point in different directions, as shown in the figure. What is true about the y-component of vector compared to the y-component of vector ?
(a) is equal in magnitude and points in the same direction to .
(b) is greater in magnitude and points in the opposite direction to .
(c) is greater in magnitude and points in the same direction to .
(d) is equal in magnitude and points in the opposite direction to .
Practice 2.2.15

Vectors and have equal magnitudes but point in different directions, as shown in the figure. If you add + , what is true about the resultant vector = + ?
(a) The resultant points in the positive y-direction and has a magnitude equal to A + B.
(b) The resultant points in the positive x-direction and has a magnitude less than A + B.
(c) The resultant points in the positive x-direction and has a magnitude equal to A + B.
(d) The resultant points in the positive y-direction and has a magnitude less than A + B.
(e) The resultant points in the negative x-direction and has a magnitude less than A + B.

Discuss!

Consider how you would answer this question. Then bring this to class for a group discussion.

An ant crawling on a table undergoes three consecutive displacements:

Find the magnitude and direction of the resultant displacement.