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Q4:
Show that the points $(2, 3, 4)$, $(-1, -2, 1)$, $(5, 8, 7)$ are collinear.
Solution :
Step 1: Establishing the Theoretical Framework
To prove that three points $A$, $B$, and $C$ are collinear, we must demonstrate that they lie on the same straight line. In three-dimensional coordinate geometry, three points are collinear if the direction ratios of the line segments joining them are proportional. Alternatively, one can show that the distance between the two furthest points is equal to the sum of the distances between the adjacent segments [By the Triangle Inequality Theorem, if $d(A, B) + d(B, C) = d(A, C)$, the points are collinear].
Step 2: Calculating Direction Ratios
Let the given points be:
- $A = (2, 3, 4)$
- $B = (-1, -2, 1)$
- $C = (5, 8, 7)$
The direction ratios $(a, b, c)$ of a line segment connecting $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are given by $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$.
For vector $\vec{AB}$:
$a_1 = -1 - 2 = -3$
$b_1 = -2 - 3 = -5$
$c_1 = 1 - 4 = -3$
Direction ratios of $AB = (-3, -5, -3)$.
For vector $\vec{BC}$:
$a_2 = 5 - (-1) = 6$
$b_2 = 8 - (-2) = 10$
$c_2 = 7 - 1 = 6$
Direction ratios of $BC = (6, 10, 6)$.
Step 3: Verification of Proportionality
Two lines are parallel (or collinear if they share a common point) if their direction ratios are proportional [Condition of Collinearity: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$].
Taking the ratio of components of $\vec{AB}$ and $\vec{BC}$:
$\frac{a_1}{a_2} = \frac{-3}{6} = -\frac{1}{2}$
$\frac{b_1}{b_2} = \frac{-5}{10} = -\frac{1}{2}$
$\frac{c_1}{c_2} = \frac{-3}{6} = -\frac{1}{2}$
Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = -\frac{1}{2}$, the direction ratios are proportional. Because the line segments $AB$ and $BC$ share a common point $B$, the points $A$, $B$, and $C$ must lie on the same straight line.
Step 4: Spatial Representation
Final Solution: Since the direction ratios of the segments $AB$ and $BC$ are proportional and the segments share the common point $B$, the points $(2, 3, 4)$, $(-1, -2, 1)$, and $(5, 8, 7)$ are proved to be collinear.
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.
- Q3: If a line has the direction ratios $-18$, $12$, $-4$, then what are its direction cosines ?
- 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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