Vector Basics

Learning Outcomes

  • Describe a plane vector, using correct notation.
  • Perform basic vector operations (scalar multiplication, addition, subtraction).

Definition


A vector is a quantity that has both magnitude and direction.

Vector Representation

A vector in a plane is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. The length of the line segment represents its magnitude. We use the notation [latex]\bf{||v||}[/latex] to denote the magnitude of the vector [latex]\bf{v}[/latex]. A vector with an initial point and terminal point that are the same is called the zero vector, denoted [latex]\bf{0}[/latex]. The zero vector is the only vector without a direction, and by convention can be considered to have any direction convenient to the problem at hand.

Vectors with the same magnitude and direction are called equivalent vectors. We treat equivalent vectors as equal, even if they have different initial points. Thus, if [latex]\bf{v}[/latex] and [latex]\bf{w}[/latex] are equivalent, we write

[latex] \bf{v = w}[/latex]

Definition


Vectors are said to be equivalent vectors if they have the same magnitude and direction.

The arrows in Figure 1(b) are equivalent. Each arrow has the same length and direction. A closely related concept is the idea of parallel vectors. Two vectors are said to be parallel if they have the same or opposite directions. We explore this idea in more detail later in the chapter. A vector is defined by its magnitude and direction, regardless of where its initial point is located.

This figure has two images. The first is labeled “a” and has a line segment representing vector v. The line segment begins at the initial point and goes to the terminal point. There is an arrowhead at the terminal point. The second image is labeled “b” and is five vectors, each labeled v sub 1, v sub 2, v sub 3, v sub 4, v sub 5. They all are pointing in the same direction and have the same length.

Figure 1 (a) A vector is represented by a directed line segment from its initial point to its terminal point. (b) Vectors [latex]{\bf{v}}_{1}[/latex] through [latex]{\bf{v}}_{5}[/latex] are equivalent.

The use of boldface, lowercase letters to name vectors is a common representation in print, but there are alternative notations. When writing the name of a vector by hand, for example, it is easier to sketch an arrow over the variable than to simulate boldface type: [latex]\overrightarrow{v}[/latex]. When a vector has initial point [latex]P[/latex] and terminal point [latex]Q[/latex], the notation [latex]\overrightarrow{PQ}[/latex] is useful because it indicates the direction and location of the vector.

Example: Sketching Vectors

Sketch a vector in the plane from initial point [latex]P(1,1)[/latex] to terminal point [latex]Q(8,5)[/latex].

Try It

Sketch the vector [latex]\overrightarrow{ST}[/latex] where [latex]S[/latex] is point [latex](3,−1)[/latex] and [latex]T[/latex] is point [latex](−2,3)[/latex].

Combining Vectors

Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.

A second example that involves vectors is a quarterback throwing a football. The quarterback does not throw the ball parallel to the ground; instead, he aims up into the air. The velocity of his throw can be represented by a vector. If we know how hard he throws the ball (magnitude—in this case, speed), and the angle (direction), we can tell how far the ball will travel down the field.

A real number is often called a scalar in mathematics and physics. Unlike vectors, scalars are generally considered to have a magnitude only, but no direction. Multiplying a vector by a scalar changes the vector’s magnitude. This is called scalar multiplication. Note that changing the magnitude of a vector does not indicate a change in its direction. For example, wind blowing from north to south might increase or decrease in speed while maintaining its direction from north to south.

Definition


The product [latex]\bf{kv}[/latex] of a vector v and a scalar k is a vector with a magnitude that is [latex]|k|[/latex] times the magnitude of [latex]\bf{v}[/latex], and with a direction that is the same as the direction of [latex]\bf{v}[/latex] if [latex]k>0[/latex], and opposite the direction of [latex]\bf{v}[/latex] if [latex]k<0[/latex]. This is called scalar multiplication. If [latex]k=0[/latex] or [latex]\bf{v=0}[/latex], then [latex]k\bf{v=0}[/latex].

As you might expect, if [latex]k=−1[/latex], we denote the product [latex]k\bf{v}[/latex] as

[latex] k\bf{v}[/latex] = [latex](-1)[/latex][latex]\bf{v}[/latex] [latex]= \bf{-v} [/latex]
Note that [latex]\bf{-v} [/latex] has the same magnitude as [latex]\bf{v}[/latex], but has the opposite direction (Figure 3).

This graphic has 4 figures. The first figure is a vector labeled “v.” The second figure is a vector twice as long as the first vector and is labeled “2 v.” The third figure is half as long as the first and is labeled “1/2 v.” The fourth figure is a vector in the opposite direction as the first. It is labeled “-v.”

Figure 4. (a) The original vector [latex]{\bf{v}}[/latex] has length [latex]n[/latex] units. (b) The length of [latex]2{\bf{v}}[/latex] equals [latex]2n[/latex] units. (c) The length of [latex]{\bf{v}}{/}2[/latex] is [latex]n{/}2[/latex] units. (d) The vectors [latex]{\bf{v}}[/latex] and [latex]-{\bf{v}}[/latex] have the same length but opposite directions.

Another operation we can perform on vectors is to add them together in vector addition, but because each vector may have its own direction, the process is different from adding two numbers. The most common graphical method for adding two vectors is to place the initial point of the second vector at the terminal point of the first, as in Figure 4(a). To see why this makes sense, suppose, for example, that both vectors represent displacement. If an object moves first from the initial point to the terminal point of vector [latex]\bf{v}[/latex], then from the initial point to the terminal point of vector [latex]\bf{w}[/latex], the overall displacement is the same as if the object had made just one movement from the initial point to the terminal point of the vector [latex]\bf{v + w}[/latex]. For obvious reasons, this approach is called the triangle method. Notice that if we had switched the order, so that [latex]\bf{w}[/latex] was our first vector and [latex]\bf{v}[/latex] was our second vector, we would have ended up in the same place. (Again, see Figure 4(a).) Thus, [latex]\bf{v + w = w + v}[/latex].

A second method for adding vectors is called the parallelogram method. With this method, we place the two vectors so they have the same initial point, and then we draw a parallelogram with the vectors as two adjacent sides, as in Figure 4(b). The length of the diagonal of the parallelogram is the sum. Comparing Figure 4(b) and Figure 4(a), we can see that we get the same answer using either method. The vector [latex]\bf{v + w}[/latex] is called the vector sum.

Definition


The sum of two vectors [latex]\bf{v}[/latex] and [latex]\bf{w}[/latex] can be constructed graphically by placing the initial point of [latex]\bf{w}[/latex] at the terminal point of [latex]\bf{v}[/latex]. Then, the vector sum, [latex]\bf{v + w}[/latex], is the vector with an initial point that coincides with the initial point of [latex]\bf{v}[/latex] and has a terminal point that coincides with the terminal point of [latex]\bf{w}[/latex]. This operation is known as vector addition.

This image has two figures. The first has two vectors, v and w with the same initial point. A parallelogram is formed by sketching broken lines parallel to the two vectors. A diagonal line is drawn from the same initial point to the opposite corner. It is labeled “v + w.” The second has two vectors, v and w. Vector v begins at the terminal point of vector w. A parallelogram is formed by sketching broken lines parallel to the two vectors. A diagonal line is drawn from the same initial point as vector w to the opposite corner. It is labeled “v + w.”

Figure 5. (a) When adding vectors by the triangle method, the initial point of [latex]{\bf{w}}[/latex] is the terminal point of [latex]{\bf{v}}[/latex]. (b) When adding vectors by the parallelogram method, the vectors [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] have the same initial point.

It is also appropriate here to discuss vector subtraction. We define [latex]\bf{v - w}[/latex] as [latex]\bf{v + (-w) = v}[/latex] [latex]+ (-1)[/latex][latex]\bf{w}[/latex]. The vector [latex]\bf{v - w}[/latex] is called the vector difference. Graphically, the vector [latex]\bf{v - w}[/latex] is depicted by drawing a vector from the terminal point of [latex]\bf{w}[/latex] to the terminal point of [latex]\bf{v}[/latex] (Figure 5).

This image has two figures. The first figure has two vectors, one labeled “v” and the other labeled “w.” Both vectors have the same initial point. A third vector is drawn between the terminal points of v and w. It is labeled “v – w.” The second figure has two vectors, one labeled “v” and the other labeled “-w.” The vector “-w” has its initial point at the terminal point of “v.” A parallelogram is created with broken lines where “v” is the diagonal and “w” is the top side.

Figure 6. (a) The vector difference [latex]{\bf{v-w}}[/latex] is depicted by drawing a vector from the terminal point of [latex]{\bf{w}}[/latex] to the terminal point of [latex]{\bf{v}}[/latex]. (b) The vector [latex]{\bf{v-w}}[/latex] is equivalent to the vector [latex]{\bf{v+(-w)}}[/latex].

In Figure 4(a), the initial point of [latex]\bf{v + w}[/latex] is the initial point of [latex]\bf{v}[/latex]. The terminal point of [latex]\bf{v + w}[/latex] is the terminal point of [latex]\bf{w}[/latex] . These three vectors form the sides of a triangle. It follows that the length of any one side is less than the sum of the lengths of the remaining sides. So we have

[latex]\bf{|| v + w|| \leq ||v|| + ||w||}[/latex].
This is known more generally as the triangle inequality. There is one case, however, when the resultant vector [latex]\bf{u + v}[/latex] has the same magnitude as the sum of the magnitudes of [latex]\bf{u}[/latex] and [latex]\bf{v}[/latex]. This happens only when [latex]\bf{u}[/latex] and [latex]\bf{v}[/latex] have the same direction.

Example: Combining Vectors

Given the vectors [latex]\bf{v}[/latex] and [latex]\bf{w}[/latex] shown in Figure 6, sketch the vectors.

    a. [latex]\bf{3w}[/latex]
    b. [latex]\bf{v + w}[/latex]
    c. [latex]\bf{2v - w}[/latex]
This figure has two vectors. They are vector v and vector w. They are not connected.

Figure 7. Vectors [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] lie in the same plane.

Try It

Using vectors [latex]\bf{w}[/latex] and [latex]\bf{w}[/latex] from Example: Combining Vectors, sketch the vector [latex]\bf{2w - v}[/latex].

Watch the following video to see the worked solution to the above Try IT.

You can view the transcript for “CP 2.2” here (opens in new window).

Try It