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Q2:
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Solution :
Step 1: Establishing Geometric Variables
Let the line make an angle $\alpha$ with the $x$-axis, $y$-axis, and $z$-axis. Since the problem specifies that the line makes equal angles with the coordinate axes, we define the direction angles as follows:
- Angle with $x$-axis: $\alpha$
- Angle with $y$-axis: $\alpha$
- Angle with $z$-axis: $\alpha$
[Per the definition of direction angles in 3D Euclidean space]
Step 2: Defining Direction Cosines
The direction cosines $(l, m, n)$ of a line are defined as the cosines of the angles the line makes with the positive coordinate axes:
- $l = \cos \alpha$
- $m = \cos \alpha$
- $n = \cos \alpha$
Step 3: Applying the Fundamental Identity
For any line in three-dimensional space, the sum of the squares of the direction cosines is always equal to unity [By the property of direction cosines in $\mathbb{R}^3$]:
$l^2 + m^2 + n^2 = 1$
Step 4: Algebraic Substitution and Solving
Substitute the expressions from Step 2 into the identity:
$\cos^2 \alpha + \cos^2 \alpha + \cos^2 \alpha = 1$
$3 \cos^2 \alpha = 1$
$\cos^2 \alpha = \frac{1}{3}$
Taking the square root on both sides:
$\cos \alpha = \pm \frac{1}{\sqrt{3}}$
Step 5: Visual Representation
Step 6: Determining the Set of Direction Cosines
Since $l = m = n = \cos \alpha$, we obtain two possible sets of direction cosines depending on the orientation of the line:
- Case 1: $l = \frac{1}{\sqrt{3}}, m = \frac{1}{\sqrt{3}}, n = \frac{1}{\sqrt{3}}$
- Case 2: $l = -\frac{1}{\sqrt{3}}, m = -\frac{1}{\sqrt{3}}, n = -\frac{1}{\sqrt{3}}$
Final Solution: The direction cosines of the line are $\left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$ or $\left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right)$.
More Questions from Class 11 Mathematics Introduction to Three Dimensional Geometry EXERCISE 11.1
- Q1: If a line makes angles $90^\circ$, $135^\circ$, $45^\circ$ with the $x$, $y$ and $z$-axes respectively, find its direction cosines.
- Q3: If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?
- Q4: Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) are collinear.
- Q5: Find the direction cosines of the sides of the triangle whose vertices are (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).
CBSE Solutions for Class 11 Mathematics Introduction to Three Dimensional Geometry
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