Vector — A Geometric Definition

Now that we have discussed what a vector is mathematically, let’s look at a more geometric inter- pretation of vectors. Geometrically speaking, a vector is a directed line segment that has magnitudeanddirection.

n The magnitudeof a vector is the length of the vector. A vector may have any nonnegative

length.

n Thedirectionof a vector describes which way the vector is pointing in space. Note thatdirec- tionis not exactly the same asorientation, a distinction we will re-examine in Section 10.1.

4.2.1 What Does a Vector Look Like?

Figure 4.1 shows an illustration of a vector in 2D:

It looks like an arrow, right? This is the standard way to represent a vector graphically since the two defining characteristics of a vector are captured, its magnitude and direction.

We will sometimes refer to theheadandtailof a vector. As shown in Figure 4.2, the head is the end of the vector with the arrow on it (where the vector “ends”), and the tail is the other end (where the vector “starts”):

Equation 4.1: Vector subscript notation

4.2.2 Position vs. Displacement

Whereis this vector? Actually, that is not an appropriate question. Vectors do not haveposition, only magnitude and direction. This may sound impossible, but many quantities we deal with on a daily basis have magnitude and direction, but no position. For example:

n Displacement:“Take three steps forward.”This sentence seems to be all about positions, but the actual quantity used in the sentence is a relative displacement and does not have an abso- lute position. This relative displacement consists of a magnitude (three steps) and a direction (forward), so it could be represented using a vector.

n Velocity:“I am traveling northeast at 50 MPH.”This sentence describes a quantity that has magnitude (50 MPH) and direction (northeast), but no position. The concept of “northeast at 50 MPH” can be represented using a vector.

Notice thatdisplacementandvelocityare technically different from the termsdistanceandspeed. Displacementandvelocityare vector quantities and entail a direction, whereasdistanceandspeed are scalar quantities and do not specify a direction.

Because vectors are used to express displacements and relative differences between things, they can describe relative positions: “My house is 3 blocks east of here.” However, you should not think of vectors as having absolute positions. (More on relative vs. absolute position in Section 4.3.1.) To help enforce this, when you imagine a vector, picture an arrow. Remember that the length and direction of this arrow is significant, but not the position.

Since vectors do not have a position, we can represent them on a diagram anywhere we choose, provided that the length and direction of the vector are represented correctly. We will often use this to our advantage by “sliding” the vector around into a meaningful location on a diagram.

4.2.3 Specifying Vectors

The numbers in a vector measuresigned displacementsin each dimension. For example, in 2D, we list the displacement parallel to both thex-axis and they-axis:

38

Figure 4.2: A vector has a head and a tail

TEAM

FLY

Figure 4.4 shows several 2D vectors and their values:

Notice that a vector’s position on the diagram is irrelevant. (The axes are conspicuously absent to enforce this policy, although we do assume the standard convention of +xpointing to the right and +ypointing up.) For example, there are two vectors shown in Figure 4.4 with the value [1.5, 1], but they are not in the same place on the diagram.

3D vectors are a simple extension of 2D vectors. A 3D vector contains three numbers which measure the signed displacements in thex,y, andzdirections, just as you’d expect.

4.2.4 Vectors as a Sequence of Displacements

One helpful way to think about the displacement described by a vector is to break the vector into its axially aligned components. When these axially aligned displacements are combined, they cumulatively define the displacement defined by the vector as a whole.

For example, the 3D vector [1, –3, 4] represents a single displacement, but we can visualize this displacement as moving one unit to the right, three units down, and then four units forward. (Assume our convention that +x, +y, and +zpoint right, up, and forward, respectively. Also note that we do not “turn” between steps, so “forward” is always parallel to +z.) This is illustrated in Figure 4.5 on the following page.

Figure 4.3: Vectors are specified by giving the signed displacement in each dimension

Figure 4.4: Examples of 2D vectors and their values

The order in which we perform the steps is not important. We could move four units forward, three units down, and then one unit to the right, and we would have displaced by the same total amount. The different orderings correspond to different routes along the axially aligned bounding box con- taining the vector. In Section 5.8, we will mathematically verify this geometric intuition.