A Geometric View of Vectors

A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:

  • Lower case, boldfaced type, with or without an arrow on top such as [latex]v,u,w,\stackrel{\to }{v},\stackrel{\to }{u},\stackrel{\to }{w}[/latex].
  • Given initial point [latex]P[/latex] and terminal point [latex]Q[/latex], a vector can be represented as [latex]\stackrel{\to }{PQ}[/latex]. The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
  • Given an initial point of [latex]\left(0,0\right)[/latex] and terminal point [latex]\left(a,b\right)[/latex], a vector may be represented as [latex]\langle a,b\rangle[/latex].

This last symbol [latex]\langle a,b\rangle[/latex] has special significance. It is called the standard position. The position vector has an initial point [latex]\left(0,0\right)[/latex] and a terminal point [latex]\langle a,b\rangle[/latex]. To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector [latex]\stackrel{\to }{CD}[/latex] is [latex]C\left({x}_{1},{y}_{1}\right)[/latex] and the terminal point is [latex]D\left({x}_{2},{y}_{2}\right)[/latex], then the position vector is found by calculating

[latex]\begin{array}{l}\stackrel{\to }{AB}=\langle {x}_{2}-{x}_{1},{y}_{2}-{y}_{1}\rangle \hfill \\ =\langle a,b\rangle \hfill \end{array}[/latex]

In Figure 2, we see the original vector [latex]\stackrel{\to }{CD}[/latex] and the position vector [latex]\stackrel{\to }{AB}[/latex].

Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.

Figure 2

A General Note: Properties of Vectors

A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at [latex]\left(0,0\right)[/latex] and is identified by its terminal point [latex]\langle a,b\rangle[/latex].

Example 1: Find the Position Vector

Consider the vector whose initial point is [latex]P\left(2,3\right)[/latex] and terminal point is [latex]Q\left(6,4\right)[/latex]. Find the position vector.

Solution

The position vector is found by subtracting one x-coordinate from the other x-coordinate, and one y-coordinate from the other y-coordinate. Thus

[latex]\begin{array}{l}v=\langle 6 - 2,4 - 3\rangle \hfill \\ =\langle 4,1\rangle \hfill \end{array}[/latex]

The position vector begins at [latex]\left(0,0\right)[/latex] and terminates at [latex]\left(4,1\right)[/latex]. The graphs of both vectors are shown in Figure 3.

Plot of the original vector in blue and the position vector in orange extending from the origin.

Figure 3

We see that the position vector is [latex]\langle 4,1\rangle[/latex].

Example 2: Drawing a Vector with the Given Criteria and Its Equivalent Position Vector

Find the position vector given that vector [latex]v[/latex] has an initial point at [latex]\left(-3,2\right)[/latex] and a terminal point at [latex]\left(4,5\right)[/latex], then graph both vectors in the same plane.

Solution

The position vector is found using the following calculation:

[latex]\begin{array}{l}v=\langle 4-\left(-3\right),5 - 2\rangle \hfill \\ \text{ }=\langle 7,3\rangle \hfill \end{array}[/latex]

Thus, the position vector begins at [latex]\left(0,0\right)[/latex] and terminates at [latex]\left(7,3\right)[/latex].

Plot of the two given vectors their same position vector.

Figure 4

Try It 1

Draw a vector [latex]v[/latex] that connects from the origin to the point [latex]\left(3,5\right)[/latex].

Solution