Vector Basics

In mathematics, some quantities can be completely described by a single number—like temperature, mass, or distance. These are called scalars. But other quantities need more information to be fully understood.

Consider describing movement or force. If you say “the wind is blowing at 15 mph,” that’s incomplete information. You also need to know the direction—north, southeast, etc. Similarly, when describing a force, you need both its strength and the direction it’s applied.

A quantity that has both magnitude (size) and direction is called a vector. We denote vectors using boldface letters like [latex]\mathbf{v}[/latex].

Vector Representation

You can represent a vector in a plane using a directed line segment (an arrow). The starting point is the initial point and the ending point is the terminal point. The arrow shows the vector’s direction, while the length of the line segment represents its magnitude. We write [latex]||\mathbf{v}||[/latex] to denote the magnitude of vector [latex]\mathbf{v}[/latex].

The zero vector, denoted [latex]\mathbf{0}[/latex], has its initial point and terminal point at the same location. This is the only vector without a specific direction—by convention, we can assign it any direction that’s convenient for the problem.

Equivalent vectors have the same magnitude and direction. We treat equivalent vectors as equal, even when they start at different points. If vectors [latex]\mathbf{v}[/latex] and [latex]\mathbf{w}[/latex] are equivalent, we write:

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

The arrows in Figure 1(b) show equivalent vectors—each has identical length and direction.

Two vectors are parallel if they point in the same direction or exact opposite directions. We’ll explore this concept in more detail later.

Are [latex]\bf{v}[/latex] and [latex]\bf{w}[/latex] equivalent vectors?

1. [latex]\bf{v}[/latex] has initial point [latex](3,2)[/latex] and terminal point [latex](7,2)[/latex]

   [latex]\bf{w}[/latex] has initial point [latex](1,−4)[/latex] and terminal point [latex](1,0)[/latex]

2. [latex]\bf{v}[/latex] has initial point [latex](0,0)[/latex] and terminal point [latex](1,1)[/latex]

   [latex]\bf{w}[/latex] has initial point [latex](−2,2)[/latex] and terminal point [latex](−1,3)[/latex]

Show Solution

1. The vectors are each [latex]4[/latex] units long, but they are oriented in different directions. So [latex]\bf{v}[/latex] and [latex]\bf{w}[/latex] are not equivalent (Figure 2.10).
   

   

2. Based on Figure 2.11, and using a bit of geometry, it is clear these vectors have the same length and the same direction, so [latex]\bf{v}[/latex] and [latex]\bf{w}[/latex] are equivalent.

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

Show Solution

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