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CDS 3D Geometry

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This page covers CDS 3D Geometry with complete concept notes, 38 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

3D Geometry— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

CORE CONCEPT 3D Geometry deals with points, lines, and planes in three-dimensional space. Instead of two coordinates (x, y), every point in 3D space has three coordinates: (x, y, z). Think of a room — length is x, width is y, and height is z. The point where all three axes meet is called the Origin (0, 0, 0).

2.Key Rules

Core rules you must know cold
Distance Formula

Distance between two points A(x1, y1, z1) and B(x2, y2, z2) is: AB = sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2] 2. Section Formula (Internal Division): Point dividing AB in ratio m:n internally: P = [(mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n)] 3

Midpoint Formula

Midpoint of AB = [(x1+x2)/2, (y1+y2)/2, (z1+z2)/2] 4

Direction Cosines

If a line makes angles alpha, beta, gamma with x, y, z axes respectively, then: cos(alpha) = l, cos(beta) = m, cos(gamma) = n Key Identity: l^2 + m^2 + n^2 = 1 5

Direction Ratios

Any multiples of direction cosines. If direction ratios are a, b, c then: l = a/sqrt(a^2+b^2+c^2), m = b/sqrt(a^2+b^2+c^2), n = c/sqrt(a^2+b^2+c^2) 6. Equation of a Line through point (x1, y1, z1) with direction ratios a, b, c: (x - x1)/a = (y - y1)/b = (z - z1)/c 7. Angle between two lines with direction cosines (l1,m1,n1) and (l2,m2,n2): cos(theta) = |l1*l2 + m1*m2 + n1*n2| 8.

Distance from Origin to point (x, y, z): r = sqrt(x^2 + y^2 + z^2)

3.Formula Block

Memorise — at least one formula appears in every paper
l^2 + m^2 + n^2 = 1 (Always true for direction cosines)
- If lines are perpendicular: l1*l2 + m1*m2 + n1*n2 = 0
- If lines are parallel: l1/l2 = m1/m2 = n1/n2

4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA regularly asks: (1) Finding distance between two 3D points, (2) Direction cosines of a line, (3) Verifying if two lines are perpendicular or parallel, (4) Finding coordinates of a dividing point. Most questions are direct formula-based. If you know the formula, you can solve in under 60 seconds.

5.Shortcuts

Use these to save 30–60 seconds per question
Trick 1 — Direction Cosine Check

If you get direction cosines from a problem, always verify l^2 + m^2 + n^2 = 1. If it does not equal 1, your answer is wrong. Use this as instant self-check

Trick 2 — Perpendicular Lines Shortcut

For lines with direction ratios (a1, b1, c1) and (a2, b2, c2), perpendicularity check is just: a1*a2 + b1*b2 + c1*c2 = 0. No need to find actual direction cosines.

6.Worked Example

Solve this step-by-step before moving on
1
Step 1

Write the distance formula: AB = sqrt[(4-1)^2 + (6-2)^2 + (3-3)^2]

2
Step 2

Calculate each bracket: (4-1)^2 = 9 (6-2)^2 = 16 (3-3)^2 = 0

3
Step 3

Add them: 9 + 16 + 0 = 25

4
Step 4

Take square root: AB = sqrt(25) = 5 Answer: Distance = 5 units

7.Exam Traps

Common mistakes students make — avoid these

Students forget the z-coordinate in the distance formula. In 2D, distance = sqrt[(x2-x1)^2 + (y2-y1)^2]. In 3D, you MUST add (z2-z1)^2.

Skipping this is the most common error in NDA papers.

Key Points to Remember

  • Every point in 3D space is written as (x, y, z) — three coordinates, three axes.
  • Distance formula in 3D: AB = sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2].
  • Direction cosines l, m, n always satisfy: l^2 + m^2 + n^2 = 1.
  • Two lines are perpendicular if: a1*a2 + b1*b2 + c1*c2 = 0.
  • Two lines are parallel if their direction ratios are proportional: a1/a2 = b1/b2 = c1/c2.
  • Midpoint of segment joining (x1,y1,z1) and (x2,y2,z2) is [(x1+x2)/2, (y1+y2)/2, (z1+z2)/2].
  • Equation of a line in 3D: (x-x1)/a = (y-y1)/b = (z-z1)/c (symmetric form).
  • Distance of point (x, y, z) from origin = sqrt(x^2 + y^2 + z^2).

Exam-Specific Tips

  • l^2 + m^2 + n^2 = 1 is the fundamental identity for direction cosines — always equals exactly 1.
  • If a line makes equal angles with all three axes, then l = m = n = 1/sqrt(3).
  • The angle between x-axis and y-axis in 3D is 90 degrees; axes are mutually perpendicular.
  • Direction ratios are NOT unique for a line — any scalar multiple is also valid.
  • Direction cosines ARE unique (up to sign) for a given directed line.
  • The coordinates of the centroid of a triangle with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) are [(x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3].
  • For a line with direction ratios (a, b, c), direction cosine l = a/sqrt(a^2+b^2+c^2).
  • Two lines in 3D can be parallel, intersecting, or skew (skew lines do not intersect and are not parallel).
Practice MCQs

3D Geometry — Practice Questions

38graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 38

All MCQs →
Practice 1easy

The scalar triple product of vectors **a** = **i** + 2**j** + 3**k**, **b** = 2**i** − **j** + **k**, and **c** = **i** + **j** − **k** is:

Practice 2easy

The direction cosines of a line passing through points P(1, 2, 3) and Q(4, 5, 6) are:

Practice 3easy

The equation of the plane passing through the point (2, 3, 1) and perpendicular to the vector **n** = (1, −2, 3) is:

Practice 4easy

The distance from the point (1, 2, 3) to the plane 2x + 2y + z − 9 = 0 is:

Practice 5easy

The angle between the planes x + 2y + 2z = 5 and 2x − y + 2z = 3 is:

Practice 6easy

The equation of the line passing through the point (1, 2, 3) and parallel to the vector **d** = (2, −1, 3) in parametric form is:

Practice 7easy

The direction cosines of a line passing through points A(1, 2, 3) and B(4, 5, 6) are:

Practice 8easy

The equation of the plane passing through the point (2, 3, 4) and perpendicular to the vector (1, 2, 3) is:

Practice 9easy

The distance from the point P(1, 2, 3) to the plane 2x − y + 2z − 9 = 0 is:

Practice 10easy

The angle between the planes x + 2y + 2z = 5 and 2x − y + 2z = 8 is:

Practice 11easy

The vector equation of the line passing through points A(1, 0, 2) and B(3, 4, 6) is:

Practice 12easy

The direction cosines of a line are proportional to 2, −3, 6. What are the actual direction cosines of this line?

Practice 13easy

Find the angle between the two lines with direction ratios 1, 2, 2 and 2, 3, −6.

Practice 14easy

The equation of a plane passing through the point (1, 2, 3) and perpendicular to the vector 2**i** + 3**j** − **k** is:

Practice 15easy

The distance from the point (2, 3, 4) to the plane x + 2y + 2z − 9 = 0 is:

Practice 16medium

The distance from the point P(2, 3, 4) to the line passing through A(1, 2, 3) with direction vector **d** = (1, 0, 1) is:

Practice 17medium

The equation of the plane containing the line **r** = (1, 0, 2) + t(1, 1, 1) and the point Q(2, 1, 3) is:

Practice 18medium

The equation of the plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratios 2 : 3 : 4 is:

Practice 19medium

The angle between the planes x + 2y + 2z = 5 and 2x − 3y + 6z = 7 is θ. Then cos θ equals:

Practice 20medium

The distance from the point (2, 3, 4) to the plane 2x − y + 2z − 9 = 0 is:

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60-Second Revision — 3D Geometry

  • Formula: 3D Distance = sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2] — always include z term.
  • Remember: l^2 + m^2 + n^2 = 1 always. Use this to verify your direction cosine answers instantly.
  • Shortcut: Perpendicular lines check — just compute a1*a2 + b1*b2 + c1*c2 = 0, no need for angles.
  • Shortcut: Parallel lines check — ratios a1/a2 = b1/b2 = c1/c2 must all be equal.
  • Trap: Do NOT confuse direction ratios with direction cosines — ratios can be any multiples, cosines must satisfy l^2+m^2+n^2=1.
  • Formula: Section formula for internal division — P = [(mx2+nx1)/(m+n), (my2+ny1)/(m+n), (mz2+nz1)/(m+n)].
  • Remember: If a line makes equal angles with all axes, each direction cosine = 1/sqrt(3) — common NDA trap question.
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