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Q3:
If a line has the direction ratios $-18$, $12$, $-4$, then what are its direction cosines ?
Solution :
Step 1: Identifying Given Parameters
The direction ratios of a line are given as:
- $a = -18$
- $b = 12$
- $c = -4$
[Per the definition of direction ratios, any triplet $(a, b, c)$ proportional to the direction cosines $(l, m, n)$ represents the orientation of the line.]
Step 2: Calculating the Magnitude of the Direction Ratios
To find the direction cosines, we must first determine the magnitude of the vector representing the direction ratios, denoted by $r = \sqrt{a^2 + b^2 + c^2}$.
$r = \sqrt{(-18)^2 + (12)^2 + (-4)^2}$
$r = \sqrt{324 + 144 + 16}$
$r = \sqrt{484}$
$r = 22$
Step 3: Deriving Direction Cosines
The direction cosines $(l, m, n)$ are calculated by dividing each direction ratio by the magnitude $r$:
$l = \frac{a}{r} = \frac{-18}{22} = -\frac{9}{11}$
$m = \frac{b}{r} = \frac{12}{22} = \frac{6}{11}$
$n = \frac{c}{r} = \frac{-4}{22} = -\frac{2}{11}$
[By the relation $l^2 + m^2 + n^2 = 1$, we verify: $(-\frac{9}{11})^2 + (\frac{6}{11})^2 + (-\frac{2}{11})^2 = \frac{81 + 36 + 4}{121} = \frac{121}{121} = 1$.]
Step 4: Consideration of Alternative Direction Cosines
Since the direction cosines are defined by the vector divided by its magnitude, taking the negative of the magnitude also yields a valid set of direction cosines for the same line (representing the opposite orientation):
$(l, m, n) = (\frac{9}{11}, -\frac{6}{11}, \frac{2}{11})$
Visual Representation
Final Solution: The direction cosines are $\left( -\frac{9}{11}, \frac{6}{11}, -\frac{2}{11} \right)$ or $\left( \frac{9}{11}, -\frac{6}{11}, \frac{2}{11} \right)$.
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.
- 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 12 Mathematics Three Dimensional Geometry
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