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

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.

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].

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See Figure 2. Because the vector goes from point [latex]P[/latex] to point [latex]Q[/latex], we name it [latex]\overrightarrow{PQ}[/latex].

Figure 2. The vector with initial point (1,1) and terminal point (8,5) is named [latex]\overrightarrow{PQ}[/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].

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Figure 3. The vector [latex]\overrightarrow{ST}[/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.

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

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

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.

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).

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

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.

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

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Figure 8.

Vector [latex]\bf{3w}[/latex] has the same direction as [latex]\bf{w}[/latex] and is three times as long.

Figure 9. To find [latex]{\bf{v+w}}[/latex], align the vectors at their initial points or place the initial point of one vector at the terminal point of the other. (a) The vector [latex]{\bf{v+w}}[/latex] is the diagonal of the parallelogram with sides [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] (b) The vector [latex]{\bf{v+w}}[/latex] is the third side of a triangle formed with [latex]{\bf{w}}[/latex] placed at the terminal point of [latex]{\bf{v}}[/latex].

Figure 10. To find [latex]2{\bf{v-w}}[/latex], simply add [latex]2{\bf{v+(-w)}}[/latex].

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].

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Figure 11. The graph of the combined vector [latex]\bf{2w - v}[/latex].

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