Three Dimensional Geometry – important questions for Boards 2027
CBSE Class 12 Maths
The direction cosines of the positive z-axis are (A) (1, 0, 0) (B) (0, 1, 0) (C) (0, 0, 1) (D) (−1, 0, 0)
Answer
(C) (0, 0, 1) — the z-axis makes angles 90°, 90° and 0° with the x-, y- and z-axes, respectively.
Find the shortest distance between the lines r⃗=λ(1,0,0) and r⃗=(0,1,2)+μ(0,1,0).
Answer
d = |(a₂ − a₁) · (b₁ × b₂)| / |b₁ × b₂| = |(0, 1, 2) · ((1, 0, 0) × (0, 1, 0))| / |(1, 0, 0) × (0, 1, 0)| = |(0, 1, 2) · (0, 0, 1)| / 1 ∴ d = 2
Find the direction cosines of the line joining P(1, 2, 2) and Q(3, 5, 6).
Answer
PQ = √[(3 − 1)² + (5 − 2)² + (6 − 2)²] = √(4 + 9 + 16) = √29 l = (3 − 1)/√29, m = (5 − 2)/√29, n = (6 − 2)/√29 ∴ The direction cosines are 2/√29, 3/√29, 4/√29.
A line makes angles 60°, 60° and 120° with the positive directions of the x-, y- and z-axes, respectively. Find its direction cosines.
Answer
l = cos 60° = 1/2 m = cos 60° = 1/2 n = cos 120° = −1/2 l² + m² + n² = 1/4 + 1/4 + 1/4 = 3/4 ≠ 1 ∴ The stated angles are inconsistent: no line can make all three angles.
Find the Cartesian equation of the line passing through (2, −1, 4) and having direction cosines 2/3, −1/3 and 2/3.
Answer
The Cartesian equation of a line through (x₁, y₁, z₁) with direction cosines l, m, n is (x − x₁)/l = (y − y₁)/m = (z − z₁)/n. (x₁, y₁, z₁) = (2, −1, 4), (l, m, n) = (2/3, −1/3, 2/3) (x − 2)/(2/3) = (y + 1)/(−1/3) = (z − 4)/(2/3) ∴ (x − 2)/2 = (y + 1)/(−1) = (z − 4)/2
Find the symmetric equation of the line passing through (1,−2,3) and having direction cosines 2/3,−1/3,2/3.
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(a) Write two possible sets of direction ratios for the segment joining A(1, 2, −1) and B(4, 0, 3). (3) (b) Show that the points P(1, 2, 3), Q(3, 6, 7) and R(5, 10, 11) are collinear by comparing the direction ratios of PQ and QR. (5) (c) Find a set of direction ratios for the line joining S(−2, 1, 4) and T(3, 5, 2), and state the other set obtained by reversing the order of the points. (4)
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Which statement describes two skew lines in space? (A) They intersect and lie in the same plane. (B) They are parallel and lie in the same plane. (C) They neither intersect nor are parallel, and lie in different planes. (D) They intersect at right angles.
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Show that the points A(1, 2, 3), B(3, 5, 7) and C(5, 8, 11) are collinear by comparing the direction ratios of AB and BC.
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Find the parametric equations of the line passing through (2, −1, 3) and having direction ratios 3, 2, −4.
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39 more probable questions in this chapter, every one with a full answer.
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