In this lesson, vectors and their basic components will be defined and quantified. For this lesson we will concentrate on 2-dimensional vectors. The Lesson:
A vector is a quantity which has both magnitude and direction. For example, velocity is a vector because it is both the speed at which an object is traveling and the direction. Speed is not a vector (it is called a scalar) because it is just the number associated with how fast you are traveling, and this number has nothing to do with direction. Let's Practice:
For example, a car may be traveling on a straight stretch of road at 50 mph in the direction NE. These two pieces of information define the velocity of the car. The speed is 50 mph.
Another way to look at the velocity of the car is to represent it as an arrow in the coordinate plane. A diagram is shown below. Because the direction is NE, the arrow points at a angle 45º with the x-axis. The length of the arrow is 50 and represents its magnitude, The arrow is a common way to represent a vector since it clearly shows a magnitude and direction. The arrow starts at point O, the initial point (or "head of the vector"), and ends at point P, the terminal point (or "tail of the vector"). It is common to call this vector , using an arrow above the initial and terminal points. can also be represented by one lower case letter either as v or .
Notice that we can determine the coordinates of point P from the triangle formed. Point P is . These coordinates are called the components of vector and are the rectangular coordinates of this vector. Using the magnitude and direction we form the polar coordinates of .
To summarize this diagram, we see that the magnitude (length) of vector is 50 and the direction is 45º with the positive x-axis. The vector has initial point (0, 0) and terminal point .
In the examples that follow, we generalize these results using the terms defined above, and show other ways that vectors can be named and combined.
We generalize the representation of a vector as an arrow directed from an initial point to a terminal point. Suppose that is a vector with the initial point, Q = (2, 5), and the terminal point, P = (3, 1). How do we find the magnitude and direction of this vector ?
A diagram is shown below.
The vector is shown as part of a right triangle. The legs of the triangle are 1 and 4. Using the Pythagorean Theorem, we can determine the magnitude of to be . The direction of this vector can be seen more clearly if we translate it to the origin so that O is the initial point and R is the terminal point. The angle with respect to the positive x-axis can be found from the adjacent side which is 1 and the opposite side which is 4. Thus the tangent of the reference angle is opp/adj = 4/1 = 4 and the angle can be found using or (4) on a calculator which is approximately 75.96º. We can call this direction either -75.96º or 284.04º.This direction can also be referred to as S by 14.04º E. There are numerous additional different, yet correct, ways to describe this direction.
Notice that the components (rectangular coordinates) of the vector are (1, -4). This can be found by subtracting the coordinates of Q from the coordinates of P as follows: . The vector has the same components as , so we say that . and are two different ways of describing the same vector. A vector is defined only by its magnitude and direction, NOT by the position of the arrow drawn on the coordinate plane. The polar coordinates of this vector are . Given its polar coordinates, its rectangular coordinates (1, -4) can be found from the formulas: .
- We can find the components of a vector by subtracting the coordinates of A from the coordinates of B.
- We can find the magnitude of a vector by applying the Pythagorean Theorem to its components. If , then the magnitude of written is .
- The direction of can be found by using .
- The components of can be found using .
- A vector has initial point A (2, 4) and terminal point B (-3, 5). What are the components of this vector and what are its direction and magnitude?
To find the components, we subtract the initial point coordinates from the terminal point coordinates:
. The magnitude is
. If this vector is translated so that the initial point is the origin, the terminal point is (-5, 1) in quadrant 2. The reference angle is in quadrant 2. The direction is approximately 168.7º or N by 78.7º W.
The polar coordinates of this vector are
. We note that
- Suppose a vector v = (-2, 7) and its terminal point is (5, 9). What is the initial point of this vector and what is , the magnitude of the vector? What are the polar coordinates of this vector?
We shall call the initial point of the vector (x, y).
v can be found by subtracting the initial point from the terminal point. Thus and we have:
The magnitude is found from the coordinates (-2, 7) and is given by
- -2 = 5 - x or x = 7
- 7 = 9 - y or y = 2
. The direction of v can be found by using and noting that this vector, if translated to the origin, is in quadrant 2. The direction is approximately
. This would give us polar coordinates of