Download Vectors - Fundamentals and Operations

Vectors - Fundamentals and Operations
Vectors and Direction
Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict
a vector by use of an arrow drawn to scale in a specific direction.
Such diagrams are commonly called as free-body diagrams.
The vector diagram depicts a displacement vector.
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a scale is clearly listed
a vector arrow (with arrowhead) is drawn in a specified direction.
The vector arrow has a head and a tail.
the magnitude and direction of the vector is clearly labeled. In this case, the
diagram shows the magnitude is 20 m and the direction is (30 degrees West of
North).
Conventions for Describing Directions of Vectors
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The direction of a vector is often expressed as an angle of rotation of the vector
about its "tail" from either east, west, north, or south. For example, a vector can
be said to have a direction of 40 degrees North of West (meaning a vector
pointing West has been rotated 40 degrees towards the northerly direction) of 65
degrees East of South (meaning a vector pointing South has been rotated 65
degrees towards the easterly direction).
The direction of a vector is often expressed as an counterclockwise angle of
rotation of the vector about its "tail" from due East. Using this convention, a
vector with a direction of 30 degrees is a vector which has been rotated 30
degrees in a counterclockwise direction relative to due east. A vector with a
direction of 160 degrees is a vector which has been rotated 160 degrees in a
counterclockwise direction relative to due east. A vector with a direction of 270
degrees is a vector which has been rotated 270 degrees in a counterclockwise
direction relative to due east. This is one of the most common conventions for the
direction of a vector and will be utilized throughout this unit.
Representing the Magnitude of a Vector
The magnitude of a vector in a scaled vector diagram is depicted by the length of the
arrow.
Vector Addition
Two vectors can be added together to determine the result (or resultant).
The Pythagorean Theorem
Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine
Eric's resulting displacement.
The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6
km. Later, the method of determining the direction of the vector will be discussed.
Using Trigonometry to Determine a Vector's Direction
The direction of a resultant vector can often be determined by use of trigonometric
functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA
from their course in trigonometry. SOH CAH TOA is a mnemonic which helps one
remember the meaning of the three common trigonometric functions - sine, cosine, and
tangent functions. These three functions relate an acute angle in a right triangle to the
ratio of the lengths of two of the sides of the right triangle. The sine function relates the
measure of an acute angle to the ratio of the length of the side opposite the angle to the
length of the hypotenuse. The cosine function relates the measure of an acute angle to the
ratio of the length of the side adjacent the angle to the length of the hypotenuse. The
tangent function relates the measure of an angle to the ratio of the length of the side
opposite the angle to the length of the side adjacent to the angle. The three equations
below summarize these three functions in equation form.
tan(Theta) = (5/10) = 0.5
Theta = tan-1 (0.5)
Theta = 26.6 degrees
Direction of R = 90 deg + 26.6 deg
Direction of R = 116.6 deg
tan(Theta) = (40/30) = 1.333
Theta = tan-1 (1.333)
Theta = 53.1 degrees
Direction of R = 180 deg + 53.1 deg
Direction of R = 233.1 deg
1. Given the SCALE: 1 cm = 10 m/s, determine the magnitude and direction of this
vector.
2. Given the SCALE: 1 cm = 50 km/hr, determine the magnitude and direction of this
vector.
3. Given the SCALE: 1 cm = 10 m/s, determine the magnitude and direction of this
vector.
4. Given the SCALE: 1 cm = 50 km/hr, determine the magnitude and direction of this
vector.
5. Given the SCALE: 1 cm = 10 m, represent the vector 50 m, 30-degrees by a scaled
vector diagram.
6. Given the SCALE: 1 cm = 10 m, represent the vector 60 m, 150-degrees by a scaled
vector diagram.
7. Given the SCALE: 1 cm = 20 m, represent the vector 140 m/s, 200-degrees by a scaled
vector diagram.
8. Given the SCALE: 1 cm = 15 m/s, represent the vector 120 m/s, 240-degrees by a
scaled vector diagram
9.Add the following vectors and determine the resultant.
3.0 m/s, 45 deg and 5.0 m/s, 135 deg
10. Add the following vectors and determine the resultant.
5.0 m/s, 45 deg and 2.0 m/s, 180 deg