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Q3:
If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?
Solution :
Step 1: Definition of Direction Ratios and Cosines
Given the direction ratios $(a, b, c)$ of a line as $-18, 12, -4$, the direction cosines $(l, m, n)$ are defined by the normalized components of the direction vector divided by its magnitude $r$.
The relations are defined as: $l = \frac{a}{r}, \quad m = \frac{b}{r}, \quad n = \frac{c}{r}$ where $r = \sqrt{a^2 + b^2 + c^2}$ [Per the definition of direction cosines in 3D Euclidean space].
Step 2: Calculation of the Magnitude ($r$)
Substitute the given values $a = -18, b = 12, c = -4$ into the magnitude formula:
$r = \sqrt{(-18)^2 + (12)^2 + (-4)^2}$
$r = \sqrt{324 + 144 + 16}$
$r = \sqrt{484}$
$r = 22$ [By the property of square roots where $\sqrt{484} = 22$].
Step 3: Determining the Direction Cosines
Since the direction cosines can be both positive and negative (depending on the direction of the line segment), we consider $r = \pm 22$.
For $r = 22$:
- $l = \frac{-18}{22} = -\frac{9}{11}$
- $m = \frac{12}{22} = \frac{6}{11}$
- $n = \frac{-4}{22} = -\frac{2}{11}$
For $r = -22$:
- $l = \frac{-18}{-22} = \frac{9}{11}$
- $m = \frac{12}{-22} = -\frac{6}{11}$
- $n = \frac{-4}{-22} = \frac{2}{11}$
Visual Representation of Direction Ratios
Verification via Identity
We verify using the identity $l^2 + m^2 + n^2 = 1$ [Fundamental property of direction cosines]:
$(-\frac{9}{11})^2 + (\frac{6}{11})^2 + (-\frac{2}{11})^2 = \frac{81}{121} + \frac{36}{121} + \frac{4}{121} = \frac{121}{121} = 1$.
Final Solution: The direction cosines are $\pm \left( -\frac{9}{11}, \frac{6}{11}, -\frac{2}{11} \right)$ or simply $\left( -\frac{9}{11}, \frac{6}{11}, -\frac{2}{11} \right)$ and $\left( \frac{9}{11}, -\frac{6}{11}, \frac{2}{11} \right)$.
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.
- Q2: Find the direction cosines of a line which makes equal angles with the coordinate axes.
- 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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