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

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

Direction Cosines and Direction Ratio is considered one of the most asked concept.
7 Questions around this concept.

Solve by difficulty

Let a unit vector $\widehat{\mathrm{OP}}$ make angles $\alpha ,\beta, \gamma$ with the positive directions of the co-ordinate axes $OX, OY, OZ$ respectively, where $\beta \in\left(0, \frac{\pi}{2}\right)$ . If $\widehat{\mathrm{OP}}$ is perpendicular to the plane through points (1,2,3), (2,3,4) and (1,5,7), then which one of the

following is true ?

$\alpha \in\left(0, \frac{\pi}{2}\right)$ and $\gamma \in\left(0, \frac{\pi}{2}\right)$

$\alpha \in\left(0, \frac{\pi}{2}\right)$ and $\gamma \in\left(\frac{\pi}{2}, \pi\right)$

$\alpha \in\left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in\left(\frac{\pi}{2}, \pi\right)$

$\alpha \in\left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in\left(0, \frac{\pi}{2}\right)$

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Concepts Covered - 1

Direction Cosines and Direction Ratio

Direction Cosines (DCs)

Let r be the position vector of a point P(x, y, z). Then, 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 denote by l, m and n respectively.

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

$\\\textbf{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\cos \alpha=\frac{x}{r}\;\;(r \text { stands for }|r|) \\\\\text {Similarly, from the right angled triangles } O B P \text { and } O C P, \\\text {We have, }\\\textbf{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\cos \beta=\frac{y}{r} \text { and } \cos \gamma=\frac{z}{r}\\\text {So we have the following results, }\\\\\mathit{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\cos \alpha=l=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}=\frac{x}{|\mathbf{r}|}=\frac{x}{r}\\\\\mathit{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\cos \beta=m=\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}=\frac{y}{|\mathbf{r}|}=\frac{y}{r}\\\\\mathit{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\cos \gamma=n=\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}=\frac{z}{|\mathbf{r}|}=\frac{z}{r}$

Also,

$\\\mathit{\;\;\;\;\;\;\;\;\;\;}l^2=\frac{x^2}{{x^{2}+y^{2}+z^{2}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(i)\\\mathit{\;\;\;\;\;\;\;\;\;\;}m^2=\frac{y^2}{{x^{2}+y^{2}+z^{2}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(ii)\\\mathit{\;\;\;\;\;\;\;\;\;\;}n^2=\frac{z^2}{{x^{2}+y^{2}+z^{2}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(iii)\\\\\textup{Add (i),\;(ii)\;and\;\;(iii)}\\l^2+m^2+n^2=\frac{x^2}{{x^{2}+y^{2}+z^{2}}}+\frac{y^2}{{x^{2}+y^{2}+z^{2}}}+\frac{z^2}{{x^{2}+y^{2}+z^{2}}}\\\\\Rightarrow l^2+m^2+n^2=1\\\\\Rightarrow \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$

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 $\lambda l, \lambda m, \lambda n$ 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.

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