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Q1:
If a line makes angles $90^{\circ}$, $135^{\circ}$, $45^{\circ}$ with the $x$, $y$ and $z$-axes respectively, find its direction cosines.
Solution :
Step 1: Identifying the Theoretical Framework
The direction cosines of a line in three-dimensional space are defined as the cosines of the angles ($\alpha, \beta, \gamma$) that the line makes with the positive directions of the $x$, $y$, and $z$-axes, respectively. These are denoted by $l, m,$ and $n$.
According to the definition:
- $l = \cos(\alpha)$
- $m = \cos(\beta)$
- $n = \cos(\gamma)$
Step 2: Assigning Given Angular Values
From the problem statement, the angles are given as:
- $\alpha = 90^{\circ}$
- $\beta = 135^{\circ}$
- $\gamma = 45^{\circ}$
Step 3: Calculating Direction Cosines
We calculate each direction cosine individually using trigonometric evaluations:
For $l$ (x-axis):
$l = \cos(90^{\circ}) = 0$
For $m$ (y-axis):
$m = \cos(135^{\circ}) = \cos(180^{\circ} - 45^{\circ})$
Using the trigonometric identity $\cos(180^{\circ} - \theta) = -\cos(\theta)$ [By the Supplementary Angle Theorem]:
$m = -\cos(45^{\circ}) = -\frac{1}{\sqrt{2}}$
For $n$ (z-axis):
$n = \cos(45^{\circ}) = \frac{1}{\sqrt{2}}$
Step 4: Verification of Identity
The direction cosines must satisfy the fundamental identity $l^2 + m^2 + n^2 = 1$.
$(0)^2 + (-\frac{1}{\sqrt{2}})^2 + (\frac{1}{\sqrt{2}})^2 = 0 + \frac{1}{2} + \frac{1}{2} = 1$.
[The identity holds, confirming the values are correct.]
Step 5: Geometric Visualization
Final Solution: The direction cosines of the line are $0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$.
More Questions from Class 12 Mathematics Three Dimensional Geometry EXERCISE 11.1
- Q2: Find the direction cosines of a line which makes equal angles with the coordinate axes.
- 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
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