After you're familiar with how to depict the velocity of a particle (see Chapter 4) and write the equation for a force vector (see Chapter 5), you want to start looking at how to work with those vectors through mathematics. Vectors become especially important when you work three-dimensional statics problems, and the skills I show you in this chapter introduce some of the methods for performing calculations with them.
In working with vectors, you soon discover a variety of basic operations that are similar to many of the basic mathematic operations you have used when working with scalar values. However, there are also some special rules that you need to observe. In this chapter, I introduce these operations and rules and show you how to apply them to your vector problems. I also give you a convenient list of properties you'll use with these operations throughout your statics work.
As you may have learned in your conventional math classes, addition and subtraction are among the most basic (and important) calculations that you work with. Vectors are no different; addition, subtraction, and relocation all become important skills, and that's what I cover in the following sections.
Simply put, the addition of vectors involves collecting each of the pieces of the action that are acting in a common direction and then representing them with some indicator of the direction of those pieces. This indicator can be another vector and in fact is often a unit vector (which is a special type of vector that has a magnitude of exactly 1.0) in the direction of each of the three Cartesian axes (covered in Chapter 5).
To add two vectors together, you simply add the scalar coefficients in front of each unit vector (i, j, and k) to make new scalar coefficients. Consider two vectors, P1 and P2:
and
The sum of these two is then
To further illustrate this concept, use Figure 6-1 to define the following navigation problem.
Using the construction techniques I discuss in Chapter 5, you can see that the position vector that defines the direct path from Point 1 to Point 3 can be given by
Instead of creating a position vector directly between Point 1 and Point 3, suppose you want a more roundabout path. You define a position vector A for path #1 between Point 1 and Point 2, such that
and another position vector B for path #2 from Point 2 to Point 3, such that
You can also travel a different path but still reach the same destination. That is, you can start at Point 1 and travel directly to Point 2 (along the position vector A), and then turn and travel from Point 2 to Point 3 (along the position vector B). In this case, your start point and end point would be exactly the same as the direct path. When you write this path out, you can see that the new path has the same start point and end point and is simply the sum of the individual path segments of the two legs of the trip (A and B). In mathematical terms,
Substituting in the expressions for A and B, notice that the result is the exact same vector. This simple example illustrates the concept of addition of vectors. What you may notice is that the original, C, is actually the sum of the individual paths (A and B) that are taken, and the ordering of the paths does not matter.
Tip
Check out the later section "Useful Vector Operation Identities" for some handy vector addition properties.
Subtracting vectors is basically the same operation as adding vectors (see the preceding section), only in reverse. The only difference is that you actually convert the vector being subtracted to a negative vector and then add the vectors. To create a negative vector, you just need to reverse the signs of each of the scalar coefficients; you can do so by simply multiplying each of the scalar terms by −1. For example, look at the following vector P1:
The negative vector of this vector is
If you want to subtract vector P1 from vector P2 in the preceding section, you just add the negative of vector P1 to vector P2 as shown in the following equation:
Figure 6-2 illustrates the subtraction of two vectors. Notice that the final result of the operation is an entirely different vector from the vector created by addition.
In this regard, the subtraction of a vector is similar to the subtraction of two scalar quantities: The term being subtracted is simply the addition of its negative representation.
In the earlier sections in this chapter, I illustrate the basic vector concepts as a series of simple steps: First, action #1 occurs, followed by another action #2. However, in the physical world, this sequence may or may not be the case. In fact, a vector may experience multiple actions simultaneously. Moving vectors head to tail when adding them is a quick and easy way of working with simultaneous actions. In fact, if you have a hundred simultaneous actions on an object, connecting the tail of one action to the head of another action for every action on the object helps you determine the combined response. The final combined response will be from the tail of the very first action you listed to the head of the very last action you listed.
Note
Order doesn't matter for simultaneous events. However, you can only attach a single tail of a vector to any given vector arrowhead. You can't attach the tails of multiple vectors to the head of the same vector.
For example, Figure 6-3a shows a baseball that has been struck by a bat. The baseball may experience a velocity in both the upward direction (as defined by vector B) as well as a velocity in a horizontal direction (as defined by vector A). Velocity A makes the ball move away from the batter, and velocity B makes it rise in the air. Each of these actions is independent of the other, and each may have a significantly different magnitude of action.
As I discuss in the "Adding vectors" section earlier in the chapter, you can represent the combined action on the baseball by adding the combined actions on the ball.
Figure 6-3b illustrates the case where the action B is drawn first. At the conclusion of action vector B (or at its head), the action A begins acting in its own direction. That is, at the conclusion of action B, the tail of action A begins.
Similarly, in Figure 6-3c, action A can be the first action that affects the baseball. Upon the conclusion of action A, action B begins. Thus, the tail for the B action is attached to the head of A.
Suppose you define a new vector C as being the combined action of A and B. The new vector C shown in both scenarios of Figure 6-3b and c, results in the magnitude, sense, and angle for the line of action (θ) of C being identical. If the three major properties of a vector are the same, the two vectors are actually the same. (Flip to Chapter 4 for more info on these basic vector properties.)
Mathematically speaking, adding and subtracting vectors and scalars are basically the same operation. However, the two remaining scalar operations — multiplication and division — work a little differently with vectors.
You can't directly multiply two vectors, but you do have other unique operations at your disposal, such as products. The following sections deal with two of the more popular products: dot products and cross products.
Tip
"Useful Vector Operation Identities" later in the chapter shows you some properties to keep in mind as you work with products.
The dot product is a type of operation that allows you to create a projection, or the portion of one vector that acts in the same direction as a second vector; it always produces a scalar result. It just requires knowing the magnitude of the two vectors involved and the angle between their lines of action. This type of operation proves to be useful in physics calculations and for quickly determining the action of one vector along the line of action of another vector. After all, if you've gone through all of the trouble of specifying the direction of a vector in its notation, it only makes sense that you should be able to put that information to work as well.
Figure 6-4 illustrates two different vectors A and B that are oriented at some arbitrary angle, θ, between them.
The dot product of Figure 6-4's vectors is defined as follows:
In these equations,
The cross product is an operation performed on two different vectors that produces a third vector that is orthogonal (perpendicular) to each of the original vectors.
Warning
Don't confuse the cross product operator (×) with the x-style multiplication operator × you learned early in your math career. These are distinctly different operations.
The cross product proves to be very useful in calculating rotational quantities called moments, which I cover in Chapter 12.
Note
Unlike the dot product, which returns a scalar quantity, the cross product computation always returns a new vector (complete with Cartesian vector notation). Check out the preceding section for dot product details.
Figure 6-5 illustrates two different vectors A and B that are oriented at some arbitrary angle θ between them.
The cross product for Figure 6-5 is defined as follows:
where n is a normal vector to both A and B. The challenge in calculating the cross product is usually in calculating the normal vector n. If you know the direction of n, the computation isn't much more difficult than the dot product calculation in the preceding section.
Unfortunately, that same vector n is often an unknown entity. Fortunately, there is a second identity, involving a determinant (a mathematical operation that utilizes a 3-x-3 matrix of values) that is much easier to calculate. The first line of the determinant always contains the unit vectors in the direction of each of the Cartesian axes. The second line is always the scalar coefficients of each of the unit vectors for the first vector listed in the cross product. The third line is always the scalar coefficients of each of the unit vectors for the second vector listed in the cross product.
For example, say you have a vector A defined as
where Ax, Ay, and Az are scalar components in the Cartesian x-, y-, and z-directions respectively, and a vector B defined as
where Bx, By, and Bz are scalar components in the Cartesian x-, y-, and z-directions respectively. You can then calculate the cross product from the determinant by using the following setup
In this example, the second line of the 3-x-3 determinant contains the scalar coefficients of the unit vectors for vector A because it's the first vector listed in the cross product. The third (or bottom) line is made up of the scalar coefficients of the unit vectors in vector B because it's the second listed vector.
Warning
You have to assemble the contents of the determinant very carefully for this method to work. Reversing the order of A and B produces a uniquely different normal vector. That is:
Tip
Figure 6-6 is a quick illustration that I like to use to help me remember the signs on all those pesky cross products. In the three locations shown, scribble the unit vectors i, j, and k. Now, locate the two vectors you want to cross. Circle the first vector in your operation and then the second vector. The term that remains uncircled is the resulting unit vector direction of that cross product operation. Now for the cool part: If your second term is located counterclockwise from the first, the sign of the result is positive. Likewise, if the second term is located clockwise from the first term, the sign of the result is negative. If the first and second terms are both the same, the result is 0. Test it out with the equations in this section — it works!
The following equations explain the results of Figure 6-6. It shows all the combinations without requiring you to memorize the nine values below, which will hopefully help you remember them a bit more easily.
