Quick Summary
Vector Algebra introduces the concept of quantities that have both magnitude and direction. It is the language of physics (force, velocity, acceleration) and is foundational for 3D geometry and calculus in Class 12. Dot product, cross product, and the section formula are the three big tools of this chapter.
Types of Vectors
| Type | Definition | ||
|---|---|---|---|
| Zero vector | Magnitude 0; direction undefined | ||
| Unit vector | Magnitude 1; â = a⃗/\ | a⃗\ | |
| Equal vectors | Same magnitude and direction | ||
| Collinear vectors | Parallel (same or opposite direction) | ||
| Coplanar vectors | Lying in the same plane | ||
| Position vector | Vector from origin to a point |
Vector Operations
Addition: a⃗ + b⃗ using triangle law or parallelogram law.
Subtraction: a⃗ − b⃗ = a⃗ + (−b⃗)
| Scalar multiplication: k·a⃗ scales the magnitude by | k | ; direction reverses if k < 0. |
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Component Form
If a⃗ = a₁î + a₂ĵ + a₃k̂ and b⃗ = b₁î + b₂ĵ + b₃k̂:
- Magnitude: \|a⃗\| = √(a₁² + a₂² + a₃²)
- Addition: a⃗ + b⃗ = (a₁+b₁)î + (a₂+b₂)ĵ + (a₃+b₃)k̂
- Unit vector: â = a⃗ / \|a⃗\|
Section Formula
Point P dividing AB in ratio m:n:
- Internal division: P⃗ = (m·b⃗ + n·a⃗)/(m+n)
- External division: P⃗ = (m·b⃗ − n·a⃗)/(m−n)
- Midpoint: M⃗ = (a⃗ + b⃗)/2
Dot Product (Scalar Product)
| **a⃗ · b⃗ = | a⃗ | b⃗ | cos θ = a₁b₁ + a₂b₂ + a₃b₃** |
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| Property | Expression | ||||
|---|---|---|---|---|---|
| Commutative | a⃗ · b⃗ = b⃗ · a⃗ | ||||
| Perpendicular vectors | a⃗ · b⃗ = 0 | ||||
| Parallel vectors | a⃗ · b⃗ = ±\ | a⃗\ | \ | b⃗\ | |
| Self dot product | a⃗ · a⃗ = \ | a⃗\ | ² | ||
| î·î = ĵ·ĵ = k̂·k̂ | = 1 | ||||
| î·ĵ = ĵ·k̂ = k̂·î | = 0 |
| Angle between vectors: cos θ = (a⃗ · b⃗) / ( | a⃗ | b⃗ | ) |
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Cross Product (Vector Product)
| **a⃗ × b⃗ = | a⃗ | b⃗ | sin θ n̂** |
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Component form: a⃗ × b⃗ = (a₂b₃ − a₃b₂)î − (a₁b₃ − a₃b₁)ĵ + (a₁b₂ − a₂b₁)k̂
(Use the 3×3 determinant with î, ĵ, k̂ in the first row)
| Property | Expression | ||
|---|---|---|---|
| Anti-commutative | a⃗ × b⃗ = −(b⃗ × a⃗) | ||
| Parallel vectors | a⃗ × b⃗ = 0⃗ | ||
| î×ĵ = k̂; ĵ×k̂ = î; k̂×î = ĵ | (cyclic) | ||
| Area of parallelogram | \ | a⃗ × b⃗\ | |
| Area of triangle | ½\ | a⃗ × b⃗\ |
Worked Examples
Example 1: Find the unit vector of a⃗ = 3î − 4ĵ.
| a⃗ | = √(9+16) = 5. â = (3î − 4ĵ)/5 = (3/5)î − (4/5)ĵ |
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Example 2: Find the angle between a⃗ = î + ĵ and b⃗ = î − ĵ. a⃗ · b⃗ = 1−1 = 0. Since dot product = 0, θ = 90° (perpendicular).
Example 3: If a⃗ = 2î + 3ĵ − k̂ and b⃗ = î − 2ĵ + 3k̂, find a⃗ × b⃗. Using determinant: = (3×3 − (−1)(−2))î − (2×3 − (−1)×1)ĵ + (2×(−2) − 3×1)k̂ = (9−2)î − (6+1)ĵ + (−4−3)k̂ = 7î − 7ĵ − 7k̂
Example 4: P divides AB internally in ratio 2:3. A = (1,2,3), B = (6,7,8). Find P. P = (2×(6,7,8) + 3×(1,2,3)) / (2+3) = ((12+3)/5, (14+6)/5, (16+9)/5) = (3, 4, 5)
Important Questions
One Mark
- Find the magnitude of 3î − 4ĵ + 12k̂.
- State the condition for two vectors to be perpendicular.
- Write the cross product î × k̂.
- What is the area of a parallelogram with adjacent sides a⃗ and b⃗?
Short Answer (3 marks)
- If a⃗ = 2î − ĵ + 3k̂ and b⃗ = î + ĵ − 2k̂, find a⃗ · b⃗ and the angle between them.
- Find the vector joining A(1,2,3) to B(4,6,9) and its unit vector.
- Show that a⃗ = 3î + 2ĵ − k̂ and b⃗ = 2î − 3ĵ are perpendicular.
Long Answer (5 marks)
- Prove that the diagonals of a rhombus bisect each other at right angles using vectors.
- Find the area of a triangle with vertices A(1,1,1), B(2,3,0), C(4,1,2) using the cross product.
Common Mistakes
- Cross product is NOT commutative — a⃗ × b⃗ = −(b⃗ × a⃗). Never swap the order without changing the sign.
- Dot product gives a scalar; cross product gives a vector — mixing them up in a calculation leads to wrong answers.
- Forgetting that î × î = 0 (not 1) — the cross product of parallel vectors is zero.
- Section formula: substituting m and n in the wrong order — m corresponds to point B (the endpoint), n to point A.
Board Exam Tips
- "Find the angle between two vectors" is the most common 3-mark question — always use the dot product formula.
- Area of triangle and parallelogram using cross product appears as a 3-mark or 5-mark question.
- Proof questions (diagonals of rhombus/rectangle) using vectors are standard 5-mark items.
- JEE/NEET: Scalar triple product and coplanarity (from Class 12) build directly on these foundations.