Three Dimensional Geometry – AAI ATC Exam Prep

📘 About This Chapter

Three Dimensional Geometry is one of the most heavily tested topics in AAI ATC Mathematics. This chapter covers Direction Cosines, Direction Ratios, Equations of Lines in Vector & Cartesian form, Angle between Lines, Skew Lines, and Shortest Distance — concepts directly appearing in AAI ATC written papers.

Direction Cosines Direction Ratios Equation of Line Angle between Lines Skew Lines Shortest Distance

Direction Cosines & Direction Ratios of a Line

A directed line L making angles α, β, γ with the positive x, y, z axes respectively has direction cosines: l = cos α, m = cos β, n = cos γ.

📌 Key Relations

l² + m² + n² = 1 (fundamental identity)

Direction Ratios (a, b, c): Any three numbers proportional to l, m, n.

l = ±a/√(a²+b²+c²), m = ±b/√(a²+b²+c²), n = ±c/√(a²+b²+c²)

DC's of line through P(x₁,y₁,z₁) and Q(x₂,y₂,z₂):

l = (x₂−x₁)/PQ, m = (y₂−y₁)/PQ, n = (z₂−z₁)/PQ

where PQ = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]

DC's of coordinate axes: x-axis → (1,0,0) | y-axis → (0,1,0) | z-axis → (0,0,1)

Collinearity Test: If DR's of AB and BC are proportional, then A, B, C are collinear.

🖼️ Image Space
Add a 3D coordinate diagram showing a line L making angles α, β, γ with x, y, z axes respectively, with the direction cosines labelled

Example: DR's −18, 12, −4 → DC's are: −18/√(324+144+16) = −18/22 = −9/11, 12/22 = 6/11, −4/22 = −2/11

Practice MCQs – Direction Cosines & Ratios

Q1. A line makes angles 90°, 60°, 30° with x, y, z axes. Its direction cosines are:

✅ Answer: B — l = cos90° = 0, m = cos60° = 1/2, n = cos30° = √3/2. Verify: 0+1/4+3/4 = 1 ✓

Q2. The direction cosines of the line joining P(−2,4,−5) and Q(1,2,3) are (PQ = √77):

✅ Answer: A — DR's: (1−(−2), 2−4, 3−(−5)) = (3,−2,8). PQ = √(9+4+64) = √77. DC's = 3/√77, −2/√77, 8/√77.

Q3. A line makes equal angles with all three coordinate axes. The value of each direction cosine is:

✅ Answer: C — If l=m=n, then l²+m²+n²=1 → 3l²=1 → l = ±1/√3. So each DC = ±1/√3.

Equation of a Line in Space

A line in 3D space is uniquely determined either by (i) a point and a direction, or (ii) two points. There are two forms: Vector form and Cartesian form.

📌 Form 1: Through point A(position vector ā) parallel to vector b̄

Vector: r̄ = ā + λb̄ (λ is parameter)

Cartesian: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c

where (x₁,y₁,z₁) is the point and (a,b,c) are direction ratios of b̄.

📌 Form 2: Through two points A(x₁,y₁,z₁) and B(x₂,y₂,z₂)

Vector: r̄ = ā + λ(b̄ − ā)

Cartesian: (x−x₁)/(x₂−x₁) = (y−y₁)/(y₂−y₁) = (z−z₁)/(z₂−z₁)

Example: Line through (5,2,−4) parallel to 3î+2ĵ−8k̂:

Vector: r̄ = (5î+2ĵ−4k̂) + λ(3î+2ĵ−8k̂)

Cartesian: (x−5)/3 = (y−2)/2 = (z+4)/−8

🖼️ Image Space
Add a 3D diagram showing a line passing through point A with direction vector b̄, with an arbitrary point P(x,y,z) marked on it and vectors ā and r̄ labelled from origin O

Practice MCQs – Equation of a Line

Q1. The Cartesian equation of the line through (1,2,3) parallel to vector 3î+2ĵ−2k̂ is:

✅ Answer: B — Point (x₁,y₁,z₁)=(1,2,3) and DR's (a,b,c)=(3,2,−2). Cartesian form: (x−1)/3 = (y−2)/2 = (z−3)/(−2).

Q2. Convert (x−5)/3 = (y+4)/7 = (z−6)/2 to vector form. The equation is:

✅ Answer: A — The point is (5,−4,6) → position vector 5î−4ĵ+6k̂. Direction ratios 3,7,2 → b̄=3î+7ĵ+2k̂. Vector form: r̄ = ā + λb̄.

Q3. The parametric equations of the line r̄ = (î+2ĵ) + λ(3î−ĵ+2k̂) are:

✅ Answer: A — r̄ = xî+yĵ+zk̂ = (1+3λ)î+(2−λ)ĵ+(2λ)k̂. So x=1+3λ, y=2−λ, z=2λ.

Angle between Two Lines

The acute angle θ between two lines can be found using their direction ratios or direction vectors.

📌 Angle Formula (Cartesian — Direction Ratios a₁,b₁,c₁ and a₂,b₂,c₂)

cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁²+b₁²+c₁²) × √(a₂²+b₂²+c₂²))

Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0

Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂

📌 Angle Formula (Vector form — lines r̄ = ā₁+λb̄₁ and r̄ = ā₂+μb̄₂)

cos θ = |b̄₁ · b̄₂| / (|b̄₁| × |b̄₂|)

Example: Lines with DR's (3,5,4) and (1,1,2):

cos θ = |3+5+8| / (√50 × √6) = 16/(5√2 × √6) = 8√3/15

θ = cos⁻¹(8√3/15)

🖼️ Image Space
Add a diagram showing two lines L₁ and L₂ through origin O with angle θ between them, direction vectors b̄₁ and b̄₂ labelled

Practice MCQs – Angle between Lines

Q1. The condition for two lines with DR's (a₁,b₁,c₁) and (a₂,b₂,c₂) to be perpendicular is:

✅ Answer: B — For perpendicular lines, cos θ = 0, which means the dot product of direction vectors = 0, i.e. a₁a₂+b₁b₂+c₁c₂ = 0.

Q2. The angle between lines r̄ = (3î+2ĵ−4k̂)+λ(î+2ĵ+2k̂) and r̄ = (5î−2ĵ)+μ(3î+2ĵ+6k̂) is:

✅ Answer: A — b̄₁=(1,2,2), b̄₂=(3,2,6). b̄₁·b̄₂=3+4+12=19. |b̄₁|=3, |b̄₂|=7. cos θ=19/21.

Q3. Find p if lines (1−x)/3 = (7y−14)/2p = (z−3)/2 and (7−7x)/3p = (y−5)/1 = (6−z)/5 are at right angles:

✅ Answer: C — DR₁: (−3, 2p/7, 2), DR₂: (−3p, 1, −5). Perpendicularity: (−3)(−3p)+(2p/7)(1)+(2)(−5)=0 → 9p+2p/7=10 → p=70/100=7/10.

Skew Lines & Shortest Distance

Skew lines are lines in space that are neither parallel nor intersecting. They lie in different planes. The shortest distance between them is the perpendicular segment joining the two lines.

📌 Shortest Distance between Two Skew Lines (Vector Form)

For lines r̄ = ā₁ + λb̄₁ and r̄ = ā₂ + μb̄₂:

d = |(b̄₁ × b̄₂) · (ā₂ − ā₁)| / |b̄₁ × b̄₂|

📌 Shortest Distance (Cartesian Form)

For lines (x−x₁)/a₁ = (y−y₁)/b₁ = (z−z₁)/c₁ and (x−x₂)/a₂ = (y−y₂)/b₂ = (z−z₂)/c₂:

d = |det[x₂−x₁, y₂−y₁, z₂−z₁; a₁, b₁, c₁; a₂, b₂, c₂]| / √[(b₁c₂−b₂c₁)²+(c₁a₂−c₂a₁)²+(a₁b₂−a₂b₁)²]

📌 Distance between Parallel Lines r̄ = ā₁+λb̄ and r̄ = ā₂+μb̄

d = |b̄ × (ā₂ − ā₁)| / |b̄|

🖼️ Image Space
Add a 3D diagram showing two skew lines l₁ and l₂ with points S and T on them, the shortest distance PQ perpendicular to both, and the common perpendicular clearly labelled

Example: Shortest distance between r̄=(î+ĵ)+λ(2î−ĵ+k̂) and r̄=(2î+ĵ−k̂)+μ(3î−5ĵ+2k̂):

ā₂−ā₁ = î−k̂ | b̄₁×b̄₂ = 3î−ĵ−7k̂ | |b̄₁×b̄₂| = √59

d = |(3î−ĵ−7k̂)·(î−k̂)| / √59 = |3+0+7| / √59 = 10/√59

Practice MCQs – Skew Lines & Shortest Distance

Q1. For two lines to be coplanar (intersecting or parallel), which condition must hold?

✅ Answer: B — Two lines are coplanar if and only if the shortest distance between them is zero, i.e. (b̄₁×b̄₂)·(ā₂−ā₁) = 0. If also b̄₁×b̄₂=0̄, the lines are parallel.

Q2. The distance between parallel lines r̄=(î+2ĵ−4k̂)+λ(2î+3ĵ+6k̂) and r̄=(3î+3ĵ−5k̂)+μ(2î+3ĵ+6k̂) is:

✅ Answer: B — ā₂−ā₁ = 2î+ĵ−k̂. b̄=2î+3ĵ+6k̂, |b̄|=7. b̄×(ā₂−ā₁) = −9î+14ĵ−4k̂. |b̄×(ā₂−ā₁)|=√(81+196+16)=√293. d=√293/7.

Q3. The shortest distance between lines (x+1)/7 = (y+1)/(−6) = (z+1)/1 and (x−3)/1 = (y−5)/(−2) = (z−7)/1 is:

✅ Answer: C — Points: (−1,−1,−1) and (3,5,7). DR₁:(7,−6,1), DR₂:(1,−2,1). b̄₁×b̄₂=(−4,−6,−8). det numerator=116. |b̄₁×b̄₂|=√116=2√29. d=116/(2√29×√29)=116/58=2√29... (standard result is 2√29).

Master Formula Table & AAI ATC Exam Tips

All key formulas in one place — revise this before your exam!

Concept	Formula
DC identity	l² + m² + n² = 1
DC from DR (a,b,c)	l=±a/√(a²+b²+c²), m=±b/√(a²+b²+c²), n=±c/√(a²+b²+c²)
DC joining P,Q	(x₂−x₁)/PQ, (y₂−y₁)/PQ, (z₂−z₁)/PQ
Line vector form	r̄ = ā + λb̄
Line Cartesian form	(x−x₁)/a = (y−y₁)/b = (z−z₁)/c
Angle between lines	cosθ = \|b̄₁·b̄₂\| / (\|b̄₁\|\|b̄₂\|)
Perpendicular lines	a₁a₂+b₁b₂+c₁c₂ = 0
Parallel lines	a₁/a₂ = b₁/b₂ = c₁/c₂
SD skew lines	\|(b̄₁×b̄₂)·(ā₂−ā₁)\| / \|b̄₁×b̄₂\|
Distance parallel lines	\|b̄×(ā₂−ā₁)\| / \|b̄\|

⚠️ AAI ATC Exam Watch Points

1. Always take the acute angle (use |cosθ| — take the modulus).

2. If (b̄₁×b̄₂)·(ā₂−ā₁) = 0, the lines are coplanar (not necessarily intersecting — could be parallel if b̄₁×b̄₂=0 too).

3. Direction ratios of a line can be scaled by any nonzero constant — there are infinitely many sets but only one set of direction cosines (for a directed line).

4. Collinearity of points: check if DR's of any two segments formed are proportional.

🖼️ Image Space
Add a summary mind-map showing all 3D Geometry concepts in a single flowchart — Direction Cosines → Line Equation → Angle → Skew Lines → Shortest Distance

Practice MCQs – Mixed Revision

Q1. The lines x/1 = y/2 = z/3 and (x−1)/2 = (y−2)/4 = (z−3)/6 are:

✅ Answer: C — DR₂=(2,4,6)=2×(1,2,3)=2×DR₁. Ratios are proportional, so lines are parallel. Point (1,2,3) of line 2 satisfies x/1=y/2=z/3, so they are actually the same line.

Q2. The direction cosines of the x-axis are:

✅ Answer: A — The x-axis makes angles 0°, 90°, 90° with x, y, z axes. So l=cos0°=1, m=cos90°=0, n=cos90°=0. DC's = (1,0,0).

Q3. Which condition shows that lines through (1,−1,2)→(3,4,−2) and (0,3,2)→(3,5,6) are perpendicular?

✅ Answer: B — DR₁=(2,5,−4), DR₂=(3,2,4). Dot product=6+10−16=0. So a₁a₂+b₁b₂+c₁c₂=0 confirms the lines are perpendicular.

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