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Q2:
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Solution :
Step 1: Establishing the Geometric Relationship
Let the line make an angle $\alpha$ with each of the $x$, $y$, and $z$ axes. Consequently, the direction angles are $\alpha = \beta = \gamma = \alpha$.
The direction cosines of a line are defined as $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$. [Per the definition of direction cosines in 3D Euclidean space].
Step 2: Applying the Fundamental Property of Direction Cosines
For any line, the sum of the squares of its direction cosines is unity [By the fundamental identity for direction cosines]:
$l^2 + m^2 + n^2 = 1$
Substituting the condition of equal angles ($\cos \alpha = \cos \beta = \cos \gamma$):
$\cos^2 \alpha + \cos^2 \alpha + \cos^2 \alpha = 1$
$3\cos^2 \alpha = 1$
Step 3: Algebraic Derivation of Values
Solving for $\cos \alpha$:
$\cos^2 \alpha = \frac{1}{3}$
$\cos \alpha = \pm \frac{1}{\sqrt{3}}$
Since $\cos \alpha = \cos \beta = \cos \gamma$, the direction cosines are:
$(l, m, n) = \left( \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}} \right)$
Step 4: Interpretation of Results
The set of direction cosines identifies the orientation of the line in space. Given the symmetry of the problem relative to the axes, two cases emerge based on the direction of the vector:
| Orientation | Direction Cosines $(l, m, n)$ |
|---|---|
| Case 1 | $\left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$ |
| Case 2 | $\left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right)$ |
Final Solution: The direction cosines are $\pm \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$.
More Questions from Class 12 Mathematics 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 12 Mathematics Three Dimensional Geometry
Chapters in CBSE - Class 12 Mathematics
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