An airplane being pushed off course by wind and a swimmer’s movement across a moving river are both examples of vectors in action. Points in the coordinate plane describe location. Vectors, on the other hand, have no location and indicate only direction and magnitude. Vectors can describe the strength of forces like gravity or speed and direction of a ship at sea. Vectors are extremely useful in modeling complex situations in the real world.
What are other differences between vectors and points?
Properties of Vectors
A two dimensional vector is represented graphically as an arrow with a tail and a head. The head is the arrow and is also called the terminal point. When finding the vector between two points start with the terminal point and subtract the initial point (the tail).
The two defining characteristics of a vector are its magnitude and its direction. The magnitude is shown graphically by the length of the arrow and the direction is indicated by the angle that the arrow is pointing. Notice how the following vector is shown multiple times on the same coordinate plane. This emphasizes that the location on the coordinate plane does not matter and is not unique. Each representation of the vector has identical direction and magnitude.
One way to define a vector is as a line segment with a direction. Vectors are said to be equal if they have the same magnitude and the same direction. The absolute value of a vector is the same as the length of the line segment or the magnitude of the vector. Magnitude can be found by using the Pythagorean Theorem or the distance formula.
Remember that when finding the vector between two points, start with the terminal point and subtract the initial point.
a. The force of gravity.
b. The force holding the daughter in the sled to the ground.
c. The force pulling the daughter backwards down the slope.
d. The force of the father pulling the daughter up the slope.
The girl’s center of gravity is represented by the black dot. The force of gravity is the black arrow straight down. The green arrow is gravity’s effect pulling the girl down the slope. The red arrow is gravity’s effect pulling the girl straight into the slope. The blue arrow represents the force that the father is exerting as he pulls the girl up the hill.
Notice that the father’s force vector (blue) is longer than the force pulling the girl down the hill. This means that over time they will make progress and ascend the hill. Also note that the father is wasting some of his energy lifting rather than just pulling. If he could pull at an angle directly opposing the force pulling the girl down the hill, then he would be using all of his energy efficiently.
Center the force diagram from the previous question into the origin and identify the angle between each consecutive force vector.
Given the following vectors and point, compute the sum.
1. Describe what a vector is and give a real-life example of something that a vector could model.
10. A ship is traveling SSW at 13 knots. Describe this ship’s movement in a vector.
For each of the following vectors, draw the vector on a coordinate plane starting at the origin and find its magnitude.
To see the Review answers, open this PDF file and look for section 7.1.