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The Concept of a Vector

Objectives

Understand the concept of a vector.
Write a vector in component form.

A scalar is a quantity having only a magnitude. For example, the temperature of a cup of coffee is a scalar describing a physical quantity of the coffee.

A vector is a quantity having both a magnitude and a direction. Forces are vectors. A force vector directed towards the center of the Earth whose magnitude is the weight of the individual may represent the weight of the person.

In this chapter, using bold typeface will indicate a vector. On the other hand, italic typeface will indicate a scalar. For example, a is a vector, whereas a is a scalar.

Given two vectors, the vectors will only be equal if both the magnitude and direction of both vectors are equal. A vector that has the same magnitude as another vector, but is in the opposite direction, is a negative vector. A vector whose magnitude is unity is called a unit vector.

A quantity such as a force has a magnitude, as well as a direction in which it is acting. This implies that vectors may represent forces.

Geometrically, we can represent vectors with an arrow. As shown in Figure 1, the head of the arrow points in the direction of the vector, and the length of the arrow indicates the magnitude of the vector.

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1 Components of a Vector

If a coordinate system is applied to a physical system, then a vector may be decomposed in vector components that act along each coordinate direction. For example, as shown in Figure 2, the vector F is the resultant of two components in the two-dimensional Cartesian coordinate system. The vector component F_x acts along the x direction, while F_y acts along the y direction. Algebraically, the resultant F may be written as

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Now, let i be a unit vector acting along the x direction and j be a unit vector acting along the y direction. The vectors i and j are sometimes called basis vectors.

Furthermore, suppose that the magnitude of F_x is F_x and F_y is F_y , then Equation 3-1 may be written as

Equation 3-2 treats an expansion for a two-dimensional Cartesian coordinate system. Similar expansions may be obtained for a vector in a non-Cartesian coordinate system.

In general, for a three-dimensional Cartesian coordinate system, a vector F may be written as

where k is the unit vector in the z direction.

From geometry (Figure 3), it may be shown that the magnitude of a vector is given by

or the square root of the sum of the squares of the components.

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EXAMPLE

A boy is pulling on a rope that is tied to a barn door, as shown in Figure 4. The boy pulls with a force of 22 N along the direction of the rope, which is at 30° to the vertical. The rope lies in a plane. What are the components of this force?

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SOLUTION

The problem statement mentions that the 22 N are directed along the direction of the rope. Therefore, we can replace the rope with a force that is located in space in the same manner as the rope. We can place a Cartesian coordinate system where the rope is tied and use geometry to find the components of the force. Because the rope lies in a plane, there are only two components of force, say, F_x and F_y .

The component in the x direction is determined by finding the projection along the x-axis as shown in Figure 4. The components of the force along each coordinate axis form a right triangle, as shown in the figure. From trigonometry, the component along the x-axis is

where the arrow indicates the direction of the component.

The component in the y direction is determined by finding the projection along the y-axis, as shown in Figure 4. This direction is determined by the cos 30°. Therefore,

2 Direction Cosines and Vectors

Another way to write a vector mathematically is to use cosines of the angles that the vector makes with the Cartesian coordinate directions. These are the angles , , and shown in Figure 3. In the figure, notice that the angles are measured between the tail of the vector F and the positive direction of the x-, y-, and z-axes. An interesting fact about these angles is that they are never greater than 180°.

From geometry, it can be shown that the cosine of each angle is equal to the projection along the direction defined by the angle divided by the magnitude of the vector. For example, the cosine of is found as follows:

Because the projection of the vector F on the x-axis is F_x, and the magnitude of F is F this becomes

Likewise,

The direction cosines are such that the sum of their squares is equal to unity. Thus,

Equation 3-7 implies that only two direction cosines need be obtained from the given information. The third direction cosine may be calculated from the equation.

Now let C be a vector whose components are the direction cosines; that is, let

Then, by using Equation 3-5 and Equation 3-6, we have

Equation 3-9 can be rearranged for the vector F in terms of the direction cosines. That is,

Equation 3-10 implies that if the direction cosines are known, an expression for the vector may be obtained.

EXAMPLE

Solve Example 3.1 by using direction cosines.

SOLUTION

In this case, there are only two direction cosines that must be found. These are related to angles and , which the tail of the vector makes with the x- and y-axes, respectively. The angles are shown in Figure 5. Their values may be readily obtained from geometry. In fact,

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and

Then, by using Equation 3-5 and Equation 3-6, the components are obtained as

These are the same results obtained in Example 3.1.

Practice!

A car is climbing a hill that has a slope of 20°, as shown in Figure 6. The car weighs 1,800 lb. The weight of the car may be assumed to act as shown in the figure. What are the components of this force in the x and y directions for the coordinate system shown in the figure?

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Professional Success

As you may have noticed from Example 3.1 and Example 3.1, solving for the components of vectors requires extensive use of geometry and trigonometry. These topics are typically studied in high school. If they are not fresh in your mind, then this might be the time for a review. One way to do this is to take a refresher course in precalculus mathematics. Most universities offer such a course.

3 Addition of Vectors

Because many forces may act on a rigid body and since forces are vectors, to solve problems involving vectors, one must be able to find the resultant of the applied forces by adding the vectors. This may be done by using a geometric technique or by summing the components of the force vector.

Vectors may be added geometrically “by joining the vectors head to tail” or “tip to tail” and making use of geometric trigonometric relations. When using this approach, use the following sequence of steps:

Construct a sketch by rearranging the vectors so that the head of one vector touches the tail of the other. Be sure to maintain the directions of the vectors.
Draw a vector from the tail of the first vector to the head of the last vector. This is the resultant vector.
Use geometry or trigonometry to obtain the magnitude and direction of the resultant vector.

If the vectors are drawn to scale, then the direction and the magnitude of the resultant vector may be obtained graphically through direct measurement on the drawing. If direct measurement is not possible, then the unknown quantities may be obtained by using the sine or the cosine law from geometry. If the vectors form a 90° triangle, then trigonometry may be used.

The geometric method works well for two coplanar vectors, but for systems with more than two vectors, extensive calculations are required. In such cases, it is better to add the components of the vector. Of course, this requires that the vectors first be written in terms of their components, which is the reason for discussing the material in Section 3.2.1 and Section 3.2.2. When using the component summation approach, use the following steps:

Obtain the components of each vector. This may be done by finding the projection of the vector along the desired direction, as discussed in Section 3.2.1, or by finding the direction cosines and using Equation 3-10 as discussed in Section 3.2.2.
Calculate the components of the resultant vector by adding the components of the vectors along the appropriate direction.

EXAMPLE

The magnitude of a vector is the sum of its components. Show that the solutions of Example 3.1 and Example 3.2 are correct by adding the components.

SOLUTION

We add the components by placing the two vectors head to tail as shown in Figure 7. Let A be the sum of these two vectors, which makes an angle with respect to the horizontal.

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The magnitude of A, A, is obtained by using the Pythagorean theorem. That is,

which is the magnitude of the initial vector.

The angle this vector makes with the horizontal may be obtained from geometry. Since

it follows that

This means that the vector makes an angle of 30° with the vertical, as defined in the original problem statement to Example 3.1.

Practice!

For the two vectors shown in Figure 8, find the resultant vector and the angle it makes with the x-axis.

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