1. Vectors and Applications
Section 8.5
Vectors and Applications
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

2. Objectives Determine whether two vectors are equivalent.
Find the sum, or resultant, of two vectors.
Resolve a vector into its horizontal and vertical components.
Solve applied problems involving vectors.

3. Terminology
Some quantities are measured using only their magnitudes: time, length and mass use units like seconds, feet, and kilograms, respectively.
However, to measure quantities like displacement, velocity, and force, we need to describe a magnitude and a direction.
Together magnitude and direction describe a vector. Here are some examples.

4. Terminology
Displacement. An object moves a certain distance in a certain direction.
A hiker follows a trail 5 mi to the west.
Velocity. An object travels at a certain speed in a certain amount of time.
A breeze is blowing 15 mph from the northwest.
Force. A push or pull is exerted on an object in a certain direction.
A force of 200 lb is required to pull a car up a 30 degree incline.

5. Vectors
Vectors can be graphically represented by directed line segments. The length is chosen, according to some scale, to represent the magnitude of the vector, and the direction of the directed line segment represents the direction of the vector. For example,
if we let 1 cm represent 5 km/h, then a 15-km/h wind from the northwest would be represented by a directed line segment 3 cm long.

6. Vector
A vector in the plane is a directed line segment. Two vectors are equivalent if they have the same magnitude and the same direction.

7. Vectors
Consider a vector drawn from point A to point B. Point A is called the initial point of the vector, and point B is called the terminal point. Symbolic notation for this vector is (read “vector AB”). Vectors are also denoted by boldface letters such as u,
v, and w. The four vectors in the figure have the same length and the same direction. Thus they represent equivalent vectors; that is,

8. Example The vectors u, and w are shown. Show that
Solution: Find the length of each vector using the distance formula.

9. Example (cont)
Thus,
Check the slopes of each vector. If the slopes are equal, then they have the same direction.
Since the vectors have the same magnitude and the same direction,

10. Vector Addition
In general, two nonzero vectors u and v can be added geometrically by placing the initial point of v at the terminal point of u and then finding the vector that has the same initial point as u and the same terminal point as v. The sum of the two vectors is also called the resultant of the two vectors.

11. Vector Addition - Parallelogram Law
We can also describe vector addition by placing the initial points of the vectors together, completing a parallelogram, and finding the diagonal of the parallelogram, called the parallelogram law of vector addition. Vector addition is commutative. As shown in the figure on the right below, both and are represented by the same directed line segment.

12. Vectors and Forces
If two forces F1 and F2 act on an object, the combined effect is the sum, or resultant, F1 + F2 of the separate forces.

13. Example
Forces of 15 newtons and 25 newtons act on an object at right angles to each other. Find their sum, or resultant, giving the magnitude of the resultant and the angle that it makes with the larger force.
Solution:
Make a drawing.
To find the magnitude use the Pythagorean equation.

14. Example (cont) OAB is a right triangle. To find the direction use,
Using a calculator,
The resultant has a magnitude of 29.2 and makes an angle of 31º with the larger force.

15. Example
An airplane travels on a bearing of 100º at an airspeed of 190 km/h while a wind is blowing 48 km/h from 220º. Find the ground speed of the airplane and the direction of its track, or course, over the ground.
Solution:
Make a few drawings.

16. Example (cont)
thus, angle CBA = 60º and angle OCB and angle OAB have the same measure = 120º

17. Example (cont) Using the law of cosines in triangle OAB, we have
By the law of sines,

18. Example (cont) Using a calculator,
Thus,  = 11º, to the nearest degree. The ground speed of the airplane is 218 km/h, and its track is in the direction of 100º – 11º, or 89º.

19. Components
Given a vector w,we may want to find two other vectors u and v whose sum is w. The vectors u and v are called components of w and the process of finding them is called resolving, or representing, a vector into its vector components.

20. Components
When we resolve a vector, we generally look for perpendicular components. Most often,one component will be parallel to the x-axis and the other will be parallel to the y-axis. For this reason, they are often called the horizontal and vertical components of a
vector. In the figure, the vector w is resolved as the sum of u and v. The horizontal component of w is u and the vertical component is v.

21. Example
A wooden shipping crate that weighs 816 lb is placed on a loading ramp that makes an angle of 25º with the horizontal. To keep the crate from sliding, a chain is hooked to the crate and to a pole at the top of the ramp. Find the magnitude of the components of the crate’s weight (disregarding friction) perpendicular and parallel to the incline.
Solution:
Make a drawing.

22. Example (cont)
= the magnitude of the component of the crate’s weight perpendicular to the incline (force against the ramp)
= the magnitude of the component of the crate’s weight perpendicular to the incline (force against the ramp)
Angle R is given to be 25º and angle BCD = angle R = 25º