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, as shown in the figure at left.
A vector in the plane is a directed line segment. Two vectors are equivalent if they have the same magnitude and direction.
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 at left have the same length and direction. Thus they represent equivalent vectors; that is,
In the context of vectors, we use = to mean equivalent.
The length, or magnitude, of is expressed as ||. In order to determine whether vectors are equivalent, we find their magnitudes and directions.
Example 1 The vectors u, , and w are shown in the figure below. Show that u = = w.
Solution We first find the length of each vector using the distance formula:
|u| = √ = √9 + 1 = √10,
|| = √ = √9 + 1 = √10,
|w| = √ = √9 + 1 = √10.
|u| = | = |w|.
The vectors u, ,and w appear to go in the same direction so we check their slopes. If the lines that they are on all have the same slope, the vectors have the same direction.We calculate the slopes:
Since u, , and w have the same magnitude and the same direction,
u = = w.
Keep in mind that the equivalence of vectors requires only the same magnitude and the same direction—not the same location. In the illustrations at left, each of the first three pairs of vectors are not equivalent. The fourth set of vectors is an example of equivalence.
Suppose a person takes 4 steps east and then 3 steps north. He or she will then be 5 steps from the starting point in the direction shown at left. A vector 4 units long and pointing to the right represents 4 steps east and a vector 3 units long and pointing up represents 3 steps north. The sum of the two vectors is the vector 5 steps in magnitude and in the direction shown. The sum is also called the resultant of the two vectors.
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, as shown in the following figure.
The sum is the vector represented by the directed line segment from the initial point A of u to the terminal point C of v. That is, if u = and v = , then
u + v = + =
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. (See the figure on the left below.) This description of addition is sometimes called the parallelogram law of vector addition. Vector addition is commutative. As shown in the figure on the right below, both u + v and v + u are represented by the same directed line segment.
If two forces F1 and F2 act on an object, the combined effect is the sum, or resultant, F1 + F2 of the separate forces.
Example 2 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 We make a drawing — this time, a rectangle — using v or to represent the resultant. To find the magnitude, we use the Pythagorean theorem:
|v|2 = 152 + 252 Here |v| denotes the length, or magnitude, of v.
|v| = √
|v| ≈ 29,2.
To find the direction, we note that since OAB is a right triangle,
tanθ = 15/25 = 0,6.
Using a calculator, we find θ, the angle that the resultant makes with the larger force:
θ = tan- 1(0,6) ≈ 31°
The resultant has a magnitude of 29,2 and makes an angle of 31° with the larger force.
Pilots must adjust the direction of their flight when there is a crosswind. Both the wind and the aircraft velocities can be described by vectors.
Example 3 Airplane Speed and Direction. An airplane travels on a bearing of 100° at an airspeed of 190 km/h while a wind is blowing 48 km/h from 220°. Find the ground speed of the airplane and the direction of its track, or course, over the ground.
Solution We first make a drawing. The wind is represented by and the velocity vector of the airplane by . The resultant velocity vector is v, the sum of the two vectors. The angle θ between v and is called a drift angle.
Note that the measure of COA = 100° - 40° = 60°. Thus the measure of CBA is also 60° (opposite angles of a parallelogram are equal). Since the sum of all the angles of the parallelogram is 360° and COB and OAB have the same measure, each must be 120°. By the law of cosines in OAB, we have
|v|2 = 482 + 1902 - 2.48.190.cos120°
|v|2 = 47,524
|v| = 218
Thus, |v| is 218 km/h. By the law of sines in the same triangle,
48/sinθ = 218/sin120°,
sinθ = 48.sin120°/218 ≈ 0,1907
θ ≈ 11°
Thus, θ = 11°, to the nearest degree. The ground speed of the airplane is 218 km/h, and its track is in the direction of 100° - 11°, or 89°.
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.
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 below, 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.
Example 4 A vector w has a magnitude of 130 and is inclined 40° with the horizontal. Resolve the vector into horizontal and vertical components.
Solution We first make a drawing showing horizontal and vertical vectors u and v whose sum is w.
From ABC, we find |u| and |v| using the definitions of the cosine and sine functions:
cos40° = |u|/130, or |u| = 130.cos40° ≈ 100,
sin40° = |v|/130, or |v| = 130.sin40° ≈ 84.
Thus the horizontal component of w is 100 right, and the vertical component of w is 84 up.