Vector Algebra
Learn Vector Algebra with notes, examples, and practice questions.
Chapter Overview
Position vector of point P(x,y,z) is OP⃗ = xî + yĵ + zk̂ from origin O.
Vectors have both magnitude and direction — unlike scalars. They model forces, velocity, displacement, and geometric directions in space.
Class 12 covers vector algebra in 3D: operations, dot product, cross product, and scalar triple product.
Vector Basics
Magnitude
|a⃗| = √(a₁² + a₂² + a₃²)
Unit vector
â = a⃗/|a⃗|
Direction ratios
Proportional to components; direction cosines l,m,n with l²+m²+n²=1
- Equal vectors: same magnitude and direction
- Collinear: a⃗ = λb⃗ for some scalar λ
- Section formula: P divides AB in m:n → OP⃗ = (n·OA⃗ + m·OB⃗)/(m+n)
Dot Product (Scalar Product)
Definition
a⃗·b⃗ = |a⃗||b⃗|cos θ
Component form
a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃
Angle
cos θ = (a⃗·b⃗)/(|a⃗||b⃗|)
Perpendicular
a⃗·b⃗ = 0
Projection
Projection of a⃗ on b⃗ = (a⃗·b⃗)/|b⃗|
Cross Product (Vector Product)
Magnitude
|a⃗×b⃗| = |a⃗||b⃗|sin θ
Direction
Perpendicular to both — right-hand rule
Parallel
a⃗×b⃗ = 0⃗
Area
Area of parallelogram = |a⃗×b⃗|; triangle = ½|a⃗×b⃗|
- î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ (cyclic)
- Cross product anti-commutative: a⃗×b⃗ = −b⃗×a⃗
Scalar Triple Product
Definition
[a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗)
Volume
Volume of parallelepiped = |[a⃗ b⃗ c⃗]|
Coplanar
[a⃗ b⃗ c⃗] = 0
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Dot Product
Find angle between a⃗ = î+ĵ+k̂ and b⃗ = î−ĵ.
- 1a⃗·b⃗ = 1−1+0 = 0
- 2cos θ = 0 → θ = 90°
Answer
90° (perpendicular)
Example 2: Cross Product
Find |î×ĵ| and direction.
- 1|î×ĵ| = sin 90° = 1
- 2Direction ⊥ to both → k̂
Answer
1, direction k̂
Example 3: Area of Triangle
Vertices A(1,0,0), B(0,2,0), C(0,0,3). Find area.
- 1AB⃗ = (−1,2,0), AC⃗ = (−1,0,3)
- 2AB⃗×AC⃗ = (6,3,2)
- 3|cross| = √(36+9+4) = 7
- 4Area = 7/2
Answer
7/2 sq. units
Example 4: Coplanarity
Are î+ĵ, ĵ+k̂, î+k̂ coplanar?
- 1Compute scalar triple product determinant
- 2[î+ĵ, ĵ+k̂, î+k̂] = 0 → coplanar
Answer
Yes, coplanar
Key Points to Remember
- ✓ Dot → scalar; cross → vector
- ✓ Use cross product for area, dot for angle and projection
- ✓ Scalar triple product for volume and coplanarity
Exam Tips
- • Write vectors in component form before computing products
- • Use determinant formula for 3×3 scalar triple product
- • Right-hand rule for cross product direction
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Vector Algebra
- Dot product:a⃗·b⃗ = |a⃗||b⃗|cos θ
- Cross product:|a⃗×b⃗| = |a⃗||b⃗|sin θ
- |a⃗ + b⃗|² = |a⃗|² + |b⃗|² + 2a⃗·b⃗
- |a⃗×b⃗|² = |a⃗|²|b⃗|² − (a⃗·b⃗)²
Practice MCQs — Vector Algebra
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
a⃗·b⃗ = 0 means vectors are: