1 The direction ratio and direction cosines

Consider the point $P\left(4,5\right)$ and its position vector $4\underset{̲}{i}+5\underset{̲}{j}$ shown in Figure 46.

Figure 46

The direction ratio of the vector $\overrightarrow{OP}$ is defined to be 4:5. We can interpret this as stating that to move in the direction of the line $OP$ we must move 4 units in the $x$ direction for every 5 units in the $y$ direction.

The direction cosines of the vector $\overrightarrow{OP}$ are the cosines of the angles between the vector and each of the axes. Specifically, referring to Figure 46 these are

$cos\alpha \text{ and }cos\beta$

Noting that the length of $\overrightarrow{OP}$ is $\sqrt{4^2+5^2}=\sqrt{41}$ , we can write

$cos\alpha =\frac{4}{\sqrt{41}},cos\beta =\frac{5}{\sqrt{41}}.$

It is conventional to label the direction cosines as $\ell$ and $m$ so that

$\ell =\frac{4}{\sqrt{41}},m=\frac{5}{\sqrt{41}}.$

More generally we have the following result:

Example 21

Point $A$ has coordinates $\left(3,5\right)$ , and point $B$ has coordinates $\left(7,8\right)$ .

1. Write down the vector $\overrightarrow{AB}$ .
2. Find the direction ratio of the vector $\overrightarrow{AB}$ .
3. Find its direction cosines, $\ell$ and $m$ .
4. Show that ${\ell }^2+m^2=1.$

Solution

1. $\overrightarrow{AB}=\underset{̲}{b}-\underset{̲}{a}=4\underset{̲}{i}+3\underset{̲}{j}$ .
2. The direction ratio of $\overrightarrow{AB}$ is therefore 4:3.
3. The direction cosines are

   $\ell =\frac{4}{\sqrt{4^2+3^2}}=\frac{4}{5},m=\frac{3}{\sqrt{4^2+3^2}}=\frac{3}{5}$

4. ${\ell }^2+m^2={\left(\frac{4}{5}\right)}^2+{\left(\frac{3}{5}\right)}^2=\frac{16}{25}+\frac{9}{25}=\frac{25}{25}=1$

The final result in the previous Example is true in general:

Exercise

$P$ and $Q$ have coordinates $\left(-2,4\right)$ and $\left(7,8\right)$ respectively.

1. Find the direction ratio of the vector $\overrightarrow{PQ}$
2. Find the direction cosines of $\overrightarrow{PQ}$ .

Answer

1. $9:4$ ,
2. $\frac{9}{\sqrt{97}}$ , $\frac{4}{\sqrt{97}}$ .