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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:

  • $l = \cos(\alpha)$
  • $m = \cos(\beta)$
  • $n = \cos(\gamma)$
[Per the definition of direction cosines in $\mathbb{R}^3$]

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

x y z Line Vector (l, m, n)

Final Solution: The direction cosines of the line are $0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$.



CBSE Solutions for Class 12 Mathematics Three Dimensional Geometry


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