Direction Cosines 

In 3D geometry, direction cosines play a key role in understanding the orientation of a line in space. Direction Cosines are the cosines of the angles that a line makes with the positive x, y, and z axes in three-dimensional space. Represented as $l=\cos \alpha ,m=\cos \beta ,andn=\cos \gamma$, they indicate the orientation of a line relative to each coordinate axis. Together, these values form a unit vector, satisfying the identity $l^2+m^2+n^2=1$. Direction cosines are essential in vector geometry, helping describe the direction of lines, resolve vectors, and calculate angles between them. They are particularly useful in physics, engineering, and competitive exams like JEE.

1.0What Are Direction Cosines?

The direction cosines of a line are the cosines of the angles between the line and the positive x, y, and z coordinate axes.

Let a line make angles α, β, and γ with the x, y, and z axes, respectively.

Then, the direction cosines are:

• l = cos α
• m = cos β
• n = cos γ

The triple (l, m, n) is called the set of direction cosines (DCs) of the line.

2.0Key Properties of Direction Cosines

1. l² + m² + n² = 1
2. If a line is parallel to a vector a = ai + bj + ck, then:

• Direction ratios (DRs) are a, b, c
• Direction cosines are: $l=\frac{a}{\sqrt{a^2+b^2+c^2}},m=\frac{b}{\sqrt{a^2+b^2+c^2}},n=\frac{c}{\sqrt{a^2+b^2+c^2}}$

3. Direction cosines are unit vectors in the direction of the line.

3.0Extended Theory of Direction Cosines

Geometric Meaning

Imagine a line in 3D space pointing in any direction. To describe how that line is oriented, we observe the angles it makes with the three coordinate axes. The cosines of these angles are the direction cosines.

DCs vs DRs

4.0Converting Direction Ratios to Direction Cosines

Given direction ratios a, b, c, convert them into direction cosines using:

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

Relationship:

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

Always valid for any direction cosines because they form a unit vector.

Use in Line Equations

A line passing through point $P(x_0,y_0,z_0)$ with direction cosines (l, m, n):

$x=x_0+lr,y=y_0+mr,z=z_0+nr$

5.0Dot Product and Direction Cosines

Given two lines with DCs:

• Line 1: $(l_1,m_1,n_1)$
• Line 2: $(l_2,m_2,n_2)$

Then: $\cos \theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$

Angles with Axes

For vector A = ai+bj+ck:

• Angle with x-axis: $cos-1(a\mid A\mid ){\cos }^{-1}\left(\frac{a}{|A|}\right)$
• Angle with y-axis: $cos-1(b\mid A\mid ){\cos }^{-1}\left(\frac{b}{|A|}\right)$
• Angle with z-axis: $cos-1(c\mid A\mid ){\cos }^{-1}\left(\frac{c}{|A|}\right)$

6.0Applications

• Physics (forces, velocity)
• Engineering (stress analysis)
• Computer Graphics (rotation, rendering)

7.0Special Cases

• Equal angles with all axes: $(13,13,13)\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$
• Along x-axis: (1, 0, 0)
• Along y-axis: (0, 1, 0)
• Along z-axis: (0, 0, 1)

8.0Solved Examples on Direction Cosines

Example 1: Find the direction cosines of a line through origin and (2, -3, 6).

Solution: 

DRs = (2, -3, 6)
Magnitude = 7

DCs = $(27,-37,67)\left(\frac{2}{7},\frac{-3}{7},\frac{6}{7}\right)$

Example 2: A line makes equal angles with x, y, and z axes. Find its DCs.

Solution: 

$l^2=1\Rightarrow l=\frac{1}{\sqrt{3}}$
DCs = $\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$

Example 3: Find angle between lines with DCs (1/3, 2/3, 2/3) and (2/3, 2/3, 1/3).

Solution: 

$\cos \theta =\frac{2}{9}+\frac{4}{9}+\frac{2}{9}=\frac{8}{9}\Rightarrow \theta ={\cos }^{-1}\left(\frac{8}{9}\right)$

9.0Practice Questions on Direction Cosines

1. Find DCs of line joining A(1, 2, 3) and B(4, 6, 9).
2. Line has DRs 5, 12, 13. Find its DCs.
3. If $l=\frac{3}{7},m=\frac{2}{7}$, find n.
4. Prove ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1.$
5. DCs = $\left(\frac{1}{2},\frac{1}{2},\frac{\sqrt{2}}{2}\right)$: Find angle with z-axis.