Three Dimensional Geometry — Class 12 MCQs with Answers

Three Dimensional Geometry — important questions & MCQs with answers (Class 12 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Direction cosines of x-axis:

   A(1,0,0)B(0,1,0)C(0,0,1)D(1,1,1)
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   Answer: (1,0,0)

   Explanation: The x-axis makes 0° with itself and 90° with the y and z axes, so its direction cosines are cos0°,cos90°,cos90° = 1,0,0 — the mixed values in the other options belong to different axes.

2. Q2 of 6Easy

   Plane ax+by+cz=d has normal vector:

   A(b,c,a)B(d,0,0)C(a,b,c)D(0,0,d)
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   Answer: (a,b,c)

   Explanation: In ax+by+cz=d, the coefficients of x, y and z form the plane's normal vector directly, in that same order — swapping the order of the coefficients gives an incorrect direction entirely.

3. Q3 of 6Medium

   Direction cosines of line with direction ratios 1, 2, 2:

   A(1/√3, 1/√3, 1/√3)B(1, 2, 2)C(1/3, 2/3, 2/3)D(1/9, 2/9, 2/9)
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   Answer: (1/3, 2/3, 2/3)

   Explanation: Divide each direction ratio by the magnitude √(1²+2²+2²)=√9=3 to get direction cosines 1/3, 2/3, 2/3 — leaving the ratios unnormalised, as the raw numbers themselves, isn't the same as direction cosines.

4. Q4 of 6Medium

   Assertion (A): The points A(2, 3, 4), B(−1, −2, 1) and C(5, 8, 7) are collinear. Reason (R): If the direction ratios of AB and BC are proportional, then the points A, B and C are collinear.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: A is true: AB has direction ratios −3, −5, −3 and BC has 6, 10, 6, which are proportional (BC = −2 × AB). R is true: proportional ratios make AB parallel to BC, and as B is common the three points lie on one line. That is exactly the test that proves A.

5. Q5 of 6Hard

   For line (x − 1)/2 = (y − 2)/3 = (z − 3)/4 = t, find point at t = 2:

   A(5, 8, 11)B(3, 5, 7)C(1, 2, 3)D(2, 3, 4)
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   Answer: (5, 8, 11)

   Explanation: At t=2: x=1+2(2)=5, y=2+3(2)=8, z=3+4(2)=11, so the point is (5,8,11) — each coordinate uses its own direction ratio as the multiplier, not a shared one.

6. Q6 of 6Hard

   Assertion (A): If the lines (1 − x)/3 = (7y − 14)/(2p) = (z − 3)/2 and (7 − 7x)/(3p) = (y − 5)/1 = (6 − z)/5 are at right angles, then p = 70/11. Reason (R): Two lines with direction ratios a₁, b₁, c₁ and a₂, b₂, c₂ are parallel if a₁/a₂ = b₁/b₂ = c₁/c₂.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: in standard form the lines are (x − 1)/(−3) = (y − 2)/(2p/7) = (z − 3)/2 and (x − 1)/(−3p/7) = (y − 5)/1 = (z − 6)/(−5); right angles need 9p/7 + 2p/7 − 10 = 0, so 11p = 70 and p = 70/11. R is true, but it is the condition for parallel lines, while A uses the perpendicular condition a₁a₂ + b₁b₂ + c₁c₂ = 0. So R does not explain A.

Three Dimensional Geometry notes: key concepts (Class 12 Mathematics)

Coordinate geometry moves to space. The chapter defines direction cosines and ratios, writes lines in vector and Cartesian form, finds angles between lines and identifies skew lines with their shortest distance, and handles planes in normal, general and intercept form — with point-to-plane distance as a key tool.

Direction cosines

l = cosα, m = cosβ, n = cosγ, the cosines of the angles a line makes with the axes; they satisfy l² + m² + n² = 1.

Direction ratios

Any numbers proportional to the direction cosines; the DRs through (x₁,y₁,z₁) and (x₂,y₂,z₂) are the coordinate differences.

Vector equation of a line

r⃗ = a⃗ + λ b⃗ passes through point a⃗ with direction b⃗; λ is a real parameter.

Cartesian equation of a line

(x − x₁)/a = (y − y₁)/b = (z − z₁)/c, where (a, b, c) are direction ratios through (x₁, y₁, z₁).

Angle between two lines

cosθ = |b⃗₁·b⃗₂|/(|b⃗₁||b⃗₂|) from their direction vectors; the lines are perpendicular when the dot product is 0.

Skew lines

Lines in space that are neither parallel nor intersecting; they do not lie in a common plane.

Shortest distance

Between skew lines the shortest distance is measured along the common perpendicular; a zero value means the lines intersect.

Coplanar / intersecting lines

Two lines meet (or are coplanar) when the vector joining a point on each is perpendicular to their common normal b⃗₁×b⃗₂.

Equation of a plane

General form ax + by + cz + d = 0, where (a, b, c) is the plane's normal vector; normal form is x cosα + y cosβ + z cosγ = p.

Plane through points / intercepts

A plane can be fixed by three non-collinear points, or written x/a + y/b + z/c = 1 using its axis intercepts a, b, c.

Angle between line and plane

sinθ = |b⃗·n⃗|/(|b⃗||n⃗|), using the line's direction and the plane's normal; note it is sine, not cosine.

Distance of a point from a plane

|ax₀ + by₀ + cz₀ + d|/√(a² + b² + c²); the perpendicular distance from (x₀, y₀, z₀) to the plane.