2 Three-dimensional coordinate frames

The real world is three-dimensional and in order to solve many engineering problems it is necessary to develop expertise in the mathematics of three-dimensional space. An important application of vectors is their use to locate points in three dimensions. When two distinct points are known we can draw a unique straight line between them. Three distinct points which do not lie on the same line form a unique plane. Vectors can be used to describe points, lines, and planes in three dimensions. These mathematical foundations underpin much of the technology associated with computer graphics and the control of robots. In this Section we shall introduce the vector methods which underlie these applications.

Figure 28

Figure 28(a) shows a three-dimensional coordinate frame. Note that the third dimension requires the addition of a third axis, the $z$ -axis. Although these three axes are drawn in the plane of the paper you should remember that we are now thinking of three-dimensional situations. Just as in two-dimensions the $x$ and $y$ axes are perpendicular, in three dimensions the $x$ , $y$ and $z$ axes are all perpendicular to each other. We say they are mutually perpendicular. There is no reason why we could not have chosen the $z$ -axis in the opposite direction. However, it is conventional to choose the directions shown in Figure 28(a).

Any point in the three dimensional space can be defined in terms of its $x$ , $y$ and $z$ coordinates. Consider the point $P$ with coordinates $\left(a,b,c\right)$ as shown in Figure 28(b). The vector from the origin to the point $P$ is known as the position vector of $P$ , denoted $\overrightarrow{OP}$ or $\underset{̲}{r}$ . To arrive at $P$ from $O$ we can think of moving $a$ units in the $x$ direction, $b$ units in the $y$ direction and $c$ units in the $z$ direction.

A unit vector pointing in the positive direction of the $z$ -axis is denoted by $\underset{̲}{k}$ . See the Figure 28(c). Noting that $\overrightarrow{OQ}=a\underset{̲}{i}+b\underset{̲}{j}$ and that $\overrightarrow{QP}=c\underset{̲}{k}$ we can state

We conclude that the position vector of the point with coordinates $\left(a,b,c\right)$ is $\underset{̲}{r}=a\underset{̲}{i}+b\underset{̲}{j}+c\underset{̲}{k}$ . (We might, for convenience, sometimes use a subscript notation. For example we might refer to the position vector $\underset{̲}{r}$ as $\underset{̲}{r}=r_x\underset{̲}{i}+r_y\underset{̲}{j}+r_z\underset{̲}{k}$ in which ( $r_x,r_y,r_z$ ) have taken the place of ( $a,b,c$ ).)

Task!

State the position vector of the point with coordinates $\left(9,-8,6\right)$ .

Answer

The modulus of the vector $\overrightarrow{OP}$ is equal to the distance $OP$ , which can be obtained by Pythagoras’ theorem:

Task!

Find the modulus of the vector $\underset{̲}{r}=4\underset{̲}{i}+2\underset{̲}{j}+3\underset{̲}{k}$ .

Answer

Example 7

Points $A$ , $B$ and $C$ have coordinates $\left(-1,1,4\right)$ , $\left(8,0,2\right)$ and $\left(5,-2,11\right)$ respectively.

1. Find the position vectors of $A$ , $B$ and $C$ .
2. Find $\overrightarrow{AB}$ and $\overrightarrow{BC}$ .
3. Find $\left|\overrightarrow{AB}\right|$ and $\left|\overrightarrow{BC}\right|$ .

Solution

1. Denoting the position vectors of $A$ , $B$ and $C$ by $\underset{̲}{a}$ , $\underset{̲}{b}$ and $\underset{̲}{c}$ respectively, we find

   $\underset{̲}{a}=-\underset{̲}{i}+\underset{̲}{j}+4\underset{̲}{k},\underset{̲}{b}=8\underset{̲}{i}+2\underset{̲}{k},\underset{̲}{c}=5\underset{̲}{i}-2\underset{̲}{j}+11\underset{̲}{k}$

2. $\overrightarrow{AB}=\underset{̲}{b}-\underset{̲}{a}=9\underset{̲}{i}-\underset{̲}{j}-2\underset{̲}{k}$ . $\overrightarrow{BC}=\underset{̲}{c}-\underset{̲}{b}=-3\underset{̲}{i}-2\underset{̲}{j}+9\underset{̲}{k}$ .
3. $\left|\overrightarrow{AB}\right|=\sqrt{9^2+{\left(-1\right)}^2+{\left(-2\right)}^2}=\sqrt{86}$ . $\left|\overrightarrow{BC}\right|=\sqrt{{\left(-3\right)}^2+{\left(-2\right)}^2+9^2}=\sqrt{94}$ .

Exercises

1. State the position vector of the point with coordinates $\left(4,-4,3\right)$ .
2. Find the modulus of each of the following vectors.
   1. $7\underset{̲}{i}+2\underset{̲}{j}+3\underset{̲}{k}$ ,
   2. $7\underset{̲}{i}-2\underset{̲}{j}+3\underset{̲}{k}$ ,
   3. $2\underset{̲}{j}+8\underset{̲}{k}$ ,
   4. $-\underset{̲}{i}-2\underset{̲}{j}+3\underset{̲}{k}$ ,
   5. $a\underset{̲}{i}+b\underset{̲}{j}+c\underset{̲}{k}$
3. Points $P$ , $Q$ and $R$ have coordinates $\left(9,1,0\right),\left(8,-3,5\right),$ and (5,5,7) respectively.
   1. Find the position vectors $\underset{̲}{p},\underset{̲}{q},\underset{̲}{r}$ of $P$ , $Q$ and $R$ ,
   2. Find $\overrightarrow{PQ}$ and $\overrightarrow{QR}$
   3. Find $\left|\overrightarrow{PQ}\right|$ and $\left|\overrightarrow{QR}\right|$ .

Answer