Direction Cosines and Direction Ratios | JEE Main Vector & 3D Geometry Crash Course | Mathematicon

Direction Cosines and Direction Ratios: Your JEE Main Compass

Hey there, JEE aspirants! Welcome to the world of 3D geometry. This lesson on Direction Cosines (DCs) and Direction Ratios (DRs) is super important for acing those JEE Main questions. Imagine them as your compass and map for navigating 3D space! You'll find these concepts popping up in vector algebra as well, so mastering them now will set you up for success later. We'll break down the concepts, formulas, and tricks to tackle any problem that comes your way. Get ready to visualize and conquer!

1. Direction Cosines (l, m, n) of a Line

Let's start with direction cosines. Think of a line in 3D space. The direction cosines are simply the cosines of the angles that the line makes with the positive directions of the x, y, and z axes, respectively. We usually denote them as $l$ , $m$ , and $n$ .

Imagine a line $L$ passing through the origin. If $\alpha$ , $\beta$ , and $\gamma$ are the angles that line $L$ makes with the x, y, and z axes, respectively, then:

$l = \cos \alpha$
$m = \cos \beta$
$n = \cos \gamma$

So, $(l, m, n)$ gives you the direction of the line $L$ in terms of angles with the coordinate axes.

2. Property: l² + m² + n² = 1

This is a fundamental property that direction cosines always satisfy. It directly stems from the Pythagorean theorem in 3D.

Derivation:

Consider a point $P(x, y, z)$ on line $L$ such that $OP = r$ (where $O$ is the origin). Then, we can express the coordinates of $P$ as:

$x = lr$
$y = mr$
$z = nr$

Now, we know that $x^2 + y^2 + z^2 = r^2$ (distance formula in 3D). Substituting the values of $x$ , $y$ , and $z$ , we get:

$(lr)^2 + (mr)^2 + (nr)^2 = r^2$

$l^2r^2 + m^2r^2 + n^2r^2 = r^2$

Dividing both sides by $r^2$ (since $r \neq 0$ ), we arrive at:

l^2 + m^2 + n^2 = 1

This equation tells us that the sum of the squares of the direction cosines is always equal to 1. Remember this – it's a lifesaver for many problems!

3. Direction Ratios (a, b, c)

Direction ratios are numbers that are proportional to the direction cosines. If $(l, m, n)$ are the direction cosines of a line, then any set of numbers $(a, b, c)$ such that:

$\frac{a}{l} = \frac{b}{m} = \frac{c}{n} = k$

where $k$ is a non-zero constant, are called the direction ratios of the line. In simple terms, direction ratios are just a scaled version of direction cosines.

4. Relation between Direction Cosines and Direction Ratios

This is where things get practical. If you know the direction ratios, you can easily find the direction cosines, and vice-versa. This relation is crucial for solving problems.

If $(a, b, c)$ are the direction ratios, then the direction cosines $(l, m, n)$ are given by:

l = \pm \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \pm \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \pm \frac{c}{\sqrt{a^2 + b^2 + c^2}}

Explanation: The $\pm$ sign indicates that there are two possible sets of direction cosines for a given set of direction ratios, corresponding to the two possible directions of the line. The denominator $\sqrt{a^2 + b^2 + c^2}$ ensures that $l^2 + m^2 + n^2 = 1$ .

Example: Let's say the direction ratios of a line are $(1, 2, -2)$ . Then, to find the direction cosines, we first calculate $\sqrt{1^2 + 2^2 + (-2)^2} = \sqrt{9} = 3$ . Therefore, the direction cosines are $(\pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{-2}{3})$ .

5. Direction Cosines of Coordinate Axes

This is a simple but important concept. The coordinate axes themselves are lines in 3D space, and they have well-defined direction cosines.

x-axis: The x-axis makes an angle of 0° with itself and 90° with both the y and z axes. Therefore, its direction cosines are:

(1, 0, 0)

y-axis: The y-axis makes an angle of 90° with the x-axis, 0° with itself, and 90° with the z-axis. Therefore, its direction cosines are:

(0, 1, 0)

z-axis: The z-axis makes an angle of 90° with both the x and y axes and 0° with itself. Therefore, its direction cosines are:

(0, 0, 1)

6. Direction Cosines of a Line Joining Two Points

This is very practical. If you have two points in space, you can find the direction ratios (and hence direction cosines) of the line joining them.

Let $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ be two points in space. Then, the direction ratios of the line joining $A$ and $B$ are given by:

(x_2 - x_1, y_2 - y_1, z_2 - z_1)

And consequently, the direction cosines are:

$l = \pm \frac{x_2 - x_1}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}$

$m = \pm \frac{y_2 - y_1}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}$

$n = \pm \frac{z_2 - z_1}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}$

Example: Consider the points $A(1, 2, 3)$ and $B(4, 5, 6)$ . The direction ratios of the line joining $A$ and $B$ are $(4-1, 5-2, 6-3) = (3, 3, 3)$ . The magnitude is $\sqrt{3^2+3^2+3^2} = \sqrt{27} = 3\sqrt{3}$ . Thus the direction cosines are $(\pm\frac{1}{\sqrt{3}}, \pm\frac{1}{\sqrt{3}}, \pm\frac{1}{\sqrt{3}})$ .

Tip: Always remember to normalize the direction ratios to get the direction cosines (i.e., divide by the magnitude).

Common Mistake: Forgetting the

\pm

sign when calculating direction cosines from direction ratios. Remember that a line can have two directions.

JEE Trick: If a question asks for the angle between two lines with given direction cosines

(l_1, m_1, n_1)

and

(l_2, m_2, n_2)

, use the formula:

\cos \theta = |l_1l_2 + m_1m_2 + n_1n_2|

Mastering direction cosines and direction ratios is fundamental for navigating 3D geometry in JEE Main. Keep practicing, and you'll be solving these problems with ease!