Concept Core
Vectors in 2D and 3D — operations, lines, planes, and spatial geometry.
Vector Operations — The Fundamentals
A vector has magnitude and direction. For a⃗ = a₁î + a₂ĵ + a₃k̂:
|a⃗| = √(a₁² + a₂² + a₃²) (magnitude)
Unit vector: â = a⃗/|a⃗|
Dot product: a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃ = |a⃗||b⃗|cosθ
Cross product: a⃗×b⃗ = |a⃗||b⃗|sinθ n̂ (n̂ ⊥ to both)
Dot Product — Key Results
a⃗·b⃗ = 0 → perpendicular vectors (θ=90°)
a⃗·b⃗ = |a⃗||b⃗| → parallel vectors (θ=0°)
Projection of a⃗ onto b⃗: (a⃗·b⃗)/|b⃗|
Component of a⃗ along b⃗: (a⃗·b⃗)b̂ = (a⃗·b⃗)b⃗/|b⃗|²
Cross Product — Key Results
a⃗×b⃗ = 0 → parallel vectors (sinθ=0)
|a⃗×b⃗| = area of parallelogram formed by a⃗ and b⃗
Area of triangle = ½|a⃗×b⃗|
î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ (cyclic). ĵ×î=−k̂ (anti-cyclic).
Scalar Triple Product
[a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗) = determinant of matrix [a|b|c]
Volume of parallelepiped = |[a⃗ b⃗ c⃗]|
[a⃗ b⃗ c⃗] = 0 → vectors are coplanar
Volume of tetrahedron = ⅙|[a⃗ b⃗ c⃗]|
Lines in 3D — Two Forms
Vector form: r⃗ = a⃗ + λb⃗ (passes through point a, direction b)
Cartesian form: (x−x₁)/l = (y−y₁)/m = (z−z₁)/n
l, m, n are direction cosines/ratios. l² + m² + n² = 1 for direction cosines.
Angle between lines: cosθ = (l₁l₂ + m₁m₂ + n₁n₂)
Planes in 3D
General equation: ax + by + cz + d = 0 (normal vector: n⃗ = (a,b,c))
Intercept form: x/a + y/b + z/c = 1
Distance from point (x₁,y₁,z₁) to plane:
d = |ax₁+by₁+cz₁+d| / √(a²+b²+c²)
Formula Vault
All vector and 3D geometry formulas.
Magnitude
|a⃗| = √(a₁²+a₂²+a₃²)
Length of vector
Dot Product
a⃗·b⃗ = a₁b₁+a₂b₂+a₃b₃
= |a||b|cosθ; scalar result
Cross Product Magnitude
|a⃗×b⃗| = |a||b|sinθ
Area of parallelogram
Angle Between Vectors
cosθ = a⃗·b⃗ / (|a⃗||b⃗|)
Use dot product formula
Perpendicularity
a⃗·b⃗ = 0
Vectors are perpendicular
Parallelism
a⃗×b⃗ = 0⃗
Vectors are parallel
Projection of a on b
(a⃗·b⃗)/|b⃗|
Scalar projection
Scalar Triple Product
[a b c] = a⃗·(b⃗×c⃗)
= det[a|b|c]; coplanar if = 0
Line in 3D
r⃗ = a⃗ + λd⃗
a = point, d = direction
Plane Equation
ax + by + cz + d = 0
Normal = (a,b,c)
Point to Plane Distance
|ax₁+by₁+cz₁+d|/√(a²+b²+c²)
Signed/unsigned distance
Triangle Area
½|a⃗×b⃗|
a,b are two sides as vectors
Worked Examples
5 problems — dot product, cross product, projection, line, and plane.
EasyFind angle between a⃗ = î+ĵ+k̂ and b⃗ = 2î−ĵ+k̂▾
Find the angle between vectors a⃗ = î+ĵ+k̂ and b⃗ = 2î−ĵ+k̂.
1
a⃗·b⃗ = (1)(2) + (1)(−1) + (1)(1) = 2 − 1 + 1 = 2
2
|a⃗| = √3, |b⃗| = √(4+1+1) = √6
3
cosθ = 2/(√3 × √6) = 2/√18 = 2/(3√2) = √2/3
✓ θ = cos⁻¹(√2/3) ≈ 61.9°
EasyCheck if a⃗ = 2î−3ĵ+k̂ and b⃗ = 3î+2ĵ are perpendicular▾
Are vectors a⃗ = 2î−3ĵ+k̂ and b⃗ = 3î+2ĵ+0k̂ perpendicular?
1
a⃗·b⃗ = (2)(3) + (−3)(2) + (1)(0) = 6 − 6 + 0 = 0
2
Since a⃗·b⃗ = 0, the vectors are perpendicular.
✓ Yes, perpendicular (dot product = 0)
MediumFind area of triangle with vertices A(1,2,3), B(3,4,5), C(0,1,2)▾
Find the area of triangle with vertices A(1,2,3), B(3,4,5), C(0,1,2).
1
AB⃗ = (3−1, 4−2, 5−3) = (2,2,2) = 2(î+ĵ+k̂)
2
AC⃗ = (0−1, 1−2, 2−3) = (−1,−1,−1) = −(î+ĵ+k̂)
3
AB⃗ × AC⃗ = 2(î+ĵ+k̂) × (−(î+ĵ+k̂)) = −2[(î+ĵ+k̂)×(î+ĵ+k̂)] = −2(0⃗) = 0⃗
4
|AB⃗ × AC⃗| = 0 → Area = 0. The three points are collinear!
✓ Area = 0 — points A, B, C are collinear
EAPCET LevelFind distance of point (1,2,3) from the plane 2x−y+2z=5▾
Find the perpendicular distance from point P(1,2,3) to plane 2x − y + 2z = 5.
1
Rewrite: 2x − y + 2z − 5 = 0. So a=2, b=−1, c=2, d=−5.
2
Distance = |ax₁+by₁+cz₁+d| / √(a²+b²+c²)
3
= |2(1)+(−1)(2)+2(3)−5| / √(4+1+4)
4
= |2 − 2 + 6 − 5| / √9 = |1| / 3 = 1/3
✓ Distance = 1/3 units
Trap Questiona⃗·(b⃗×c⃗) = 0 means what about the vectors?▾
If a⃗·(b⃗×c⃗) = 0, students often give wrong geometric interpretation. What does it actually mean?
1
The scalar triple product [a b c] = a⃗·(b⃗×c⃗) = volume of parallelepiped.
2
If [a b c] = 0: volume = 0, meaning the three vectors lie in the same plane.
3
The trap: Students say 'a is perpendicular to b×c' (which is true geometrically) but forget the simpler interpretation: vectors are coplanar.
4
Coplanar means: all three vectors start from the same point and lie in one flat plane. This is the key EAPCET test.
✓ Vectors are coplanar (lie in the same plane)
Mistake DNA
4 vector errors that EAPCET distractors are built around.
↔️
Dot Product is Commutative, Cross Product is NOT
a⃗·b⃗ = b⃗·a⃗ (commutative). But a⃗×b⃗ = −b⃗×a⃗ (anti-commutative). Sign error in cross product order is very common.
❌ Wrong
a⃗×b⃗ = b⃗×a⃗ ✗
(cross product:
swapping reverses sign)
✓ Correct
a⃗·b⃗ = b⃗·a⃗ ✓
a⃗×b⃗ = −b⃗×a⃗ ✓
Cross: order matters
Dot product: scalar, commutative. Cross product: vector, anti-commutative. Swapping the order of cross product vectors flips the sign of the result.
📐
Direction Cosines: Using Ratios Instead of Cosines
Direction cosines l,m,n satisfy l²+m²+n²=1. Direction ratios are proportional but not normalised.
❌ Wrong
For direction ratios 1,2,2:
l=1, m=2, n=2 ✗
(not normalised;
l²+m²+n² = 9 ≠ 1)
✓ Correct
√(1²+2²+2²) = 3
l=1/3, m=2/3, n=2/3 ✓
l²+m²+n² = 1 ✓
Direction cosines are the direction ratios normalised by the magnitude. Always divide by √(a²+b²+c²) to get the actual cosines.
🔢
Cross Product: Wrong Determinant Expansion
The cross product determinant: top row is î,ĵ,k̂; second row is a₁,a₂,a₃; third row is b₁,b₂,b₃. Mixing up rows gives wrong result.
❌ Wrong
a⃗×b⃗: expanding along
a wrong row or column ✗
(common arithmetic error)
✓ Correct
a⃗×b⃗ = det|î ĵ k̂ / a₁ a₂ a₃ / b₁ b₂ b₃| ✓
Expand along first row
checking signs: + − +
The 3×3 determinant for cross product: cofactor expansion along first row with signs (+,−,+). Practice writing this out slowly — speed comes from practice.
🌐
Scalar Triple Product: Wrong Coplanarity Conclusion
[a b c] = 0 means coplanar. Students sometimes say it means perpendicular or parallel.
❌ Wrong
[a b c] = 0:
'a is perpendicular to b' ✗
(that would be a·b=0)
✓ Correct
[a b c] = 0:
→ vectors are coplanar ✓
→ parallelepiped volume = 0 ✓
[a b c] = 0 has one geometric meaning: the three vectors are coplanar (lie in one plane). Test: if any two are parallel, triple product = 0 (that's a special case of coplanar).
Chapter Intelligence
Vectors is a prerequisite for 3D geometry, and connects to Physics mechanics and Coordinate Geometry.
EAPCET Weightage (2019–2024)
Dot product & angle between vectors~8 Lines in 3D — direction cosines~6 Planes — distance, angle~5 High-Yield PYQ Patterns
Angle between vectors using dot productArea of triangle/parallelogramCoplanarity conditionDistance from point to planeLine direction cosines l²+m²+n²=1Cross product to find perpendicular vectorProjection of a onto b
Exam Strategy
- Dot product questions: compute the algebraic dot product (sum of products) and set equal to |a||b|cosθ to find angle. Fast and clean.
- For perpendicularity: just check a⃗·b⃗ = 0. No need to compute magnitudes or angles.
- Area questions: always use ½|a⃗×b⃗| where a⃗ and b⃗ are two sides of the triangle as vectors from the same vertex.
- Coplanarity: compute scalar triple product. If [a b c] = 0, coplanar. This test appears every 2–3 EAPCET papers.
- Distance from point to plane: formula |ax₁+by₁+cz₁+d|/√(a²+b²+c²). Memorise this — it's a direct formula question every year.