Direction Cosines & Direction Ratios Of A Line - Practice Questions & MCQ

Direction Cosines (DCs)

Let r be the position vector of a point P(x, y, z). Then, the direction cosines of vector r are the cosines of angles α, β, and γ (i.e. cos α, cos β, and cos γ) that the vector r makes with the positive direction of X, Y, and Z -axes respectively. Direction cosines are usually denoted by l, m, and n respectively.

From the figure, note that ΔOAP is a right-angled triangle and thus, we have

$$
\cos \alpha=\frac{x}{r}(r \text { stands for }|r|)
$$

Similarly, from the right angled triangles $O B P$ and $O C P$, We have,
$$
\cos \beta=\frac{y}{r} \text { and } \cos \gamma=\frac{z}{r}
$$

So we have the following results,
$$
\begin{aligned}
& \cos \alpha=l=\frac{x}{\sqrt{x^2+y^2+z^2}}=\frac{x}{|\mathbf{r}|}=\frac{x}{r} \\
& \cos \beta=m=\frac{y}{\sqrt{x^2+y^2+z^2}}=\frac{y}{|\mathbf{r}|}=\frac{y}{r} \\
& \cos \gamma=n=\frac{z}{\sqrt{x^2+y^2+z^2}}=\frac{z}{|\mathbf{r}|}=\frac{z}{r}
\end{aligned}
$$

Also,
$$
\begin{aligned}
& l^2=\frac{x^2}{x^2+y^2+z^2} \\
& m^2=\frac{y^2}{x^2+y^2+z^2} \\
& n^2=\frac{z^2}{x^2+y^2+z^2}
\end{aligned}
$$

Add (i), (ii) and (iii)
$$
\begin{aligned}
& l^2+m^2+n^2=\frac{x^2}{x^2+y^2+z^2}+\frac{y^2}{x^2+y^2+z^2}+\frac{z^2}{x^2+y^2+z^2} \\
& \Rightarrow l^2+m^2+n^2=1 \\
& \Rightarrow \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1
\end{aligned}
$$

The coordinates of the point P may also be expressed as (lr, mr, nr).

Direction ratios (DRs)

Direction Ratios are any set of three numbers that are proportional to the Direction cosines.

If l, m, n are DCs of a vector then  are DRs of this vector, where a can take any real value.

DRs are also denoted as a, b, and c, respectively.

A vector has only one set of DCs, but infinite sets of DRs.

Note:
The coordinates of a point equal lr, Mr, and nr, which are proportional to the direction cosines. Hence the coordinates of a point are also its DRs.

If $\vec{r}=ai+b \hat{j}+c \hat{k}$, then $a, b$ and $c$ are one of the direction ratios of the given vector. Also, if $a^2+b^2+c^2=1$, then $a, b$ and $c$ will be direction cosines of the given vector.