2 Direction ratios and cosines in three dimensions

The concepts of direction ratio and direction cosines extend naturally to three dimensions. Consider Figure 47.

Figure 47

Given a vector $\underset{̲}{r}=a\underset{̲}{i}+b\underset{̲}{j}+c\underset{̲}{k}$ its direction ratios are $a:b:c$ . This means that to move in the direction of the vector we must must move $a$ units in the $x$ direction and $b$ units in the $y$ direction for every $c$ units in the $z$ direction.

The direction cosines are the cosines of the angles between the vector and each of the axes. It is conventional to label direction cosines as $\ell$ , $m$ and $n$ and they are given by

Wee have the following general result:

Exercises

1. Points $A$ and $B$ have position vectors $\underset{̲}{a}=-3\underset{̲}{i}+2\underset{̲}{j}+7\underset{̲}{k}$ , and $\underset{̲}{b}=3\underset{̲}{i}+4\underset{̲}{j}-5\underset{̲}{k}$ respectively. Find
   1. $\overrightarrow{AB}$
   2. $\left|\overrightarrow{AB}\right|$
   3. The direction ratios of $\overrightarrow{AB}$
   4. The direction cosines $\left(\ell ,m,n\right)$ of $\overrightarrow{AB}$ .
   5. Show that ${\ell }^2+m^2+n^2=1$ .
2. Find the direction ratios, the direction cosines and the angles that the vector $\overrightarrow{OP}$ makes with each of the axes when $P$ is the point with coordinates $\left(2,4,3\right)$ .
3. A line is inclined at $60^{\circ }$ to the $x$ axis and $45^{\circ }$ to the $y$ axis. Find its inclination to the $z$ axis.

Answer