๐ CHAPTER 10: VECTOR ALGEBRA
Subject: Mathematics | Class: XII | CBSE
๐ INTRODUCTION
Vector
Quantities which have both magnitude and direction are called vector quantities or vectors.
Scalar
Quantities which have only magnitude and no direction are called scalar quantities.
๐น Representation of a Vector
A directed line segment is called a vector, denoted as \vec{AB} or \vec{a}.
- Initial point: A (starting point)
- Terminal point: B (ending point)
๐น Magnitude of a Vector
The length of the vector \vec{AB} or \vec{a} is called its magnitude.
|\vec{AB}| or |\vec{a}|
๐ก Note: Since length is never negative, |\vec{a}| < 0 has no meaning.
๐ POSITION VECTOR
Position Vector
Let O(0, 0, 0) be the origin and P(x, y, z) be a point in space.
The vector \vec{OP} or \vec{r} is called the position vector of point P with respect to O.
\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}
|\vec{r}| = √(x² + y² + z²)
๐น Direction Cosines
If ฮฑ, ฮฒ, ฮณ are the angles that \vec{OP} makes with OX, OY, OZ axes respectively, then:
l = cos ฮฑ, m = cos ฮฒ, n = cos ฮณ
These are called direction cosines.
l² + m² + n² = 1
๐ข TYPES OF VECTORS
Null Vector (Zero Vector)
A vector whose initial and terminal points coincide and magnitude is zero.
Denoted by \vec{0}.
Unit Vector
A vector of unit length (magnitude = 1).
\hat{a} = \vec{a}/|\vec{a}|
Collinear Vectors
Vectors parallel to the same line.
\vec{a} = ±ฮป\vec{b}
Coinitial Vectors
Vectors having the same initial point.
Equal Vectors
Vectors with equal magnitudes and same direction.
Negative Vector
Vector with same magnitude but opposite direction.
\vec{BA} = -\vec{AB}
➕ ADDITION OF VECTORS
๐น Triangle Law of Vector Addition
If two vectors are represented along two sides of a triangle taken in order, then their resultant is represented by the third side taken in opposite direction.
\vec{BC} + \vec{CA} = \vec{BA}
๐น Parallelogram Law of Vector Addition
If two vectors are represented along the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the parallelogram.
\vec{OA} + \vec{OC} = \vec{OB}
๐น Properties of Vector Addition
Commutative
\vec{a} + \vec{b} = \vec{b} + \vec{a}
Associative
\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}
Additive Identity
\vec{a} + \vec{0} = \vec{a}
Additive Inverse
\vec{a} + (-\vec{a}) = \vec{0}
✖ SCALAR MULTIPLICATION
Scalar Multiplication
If \vec{a} is a vector and ฮป is a scalar, then ฮป\vec{a} is a vector:
- Magnitude: |ฮป| times |\vec{a}|
- Direction: Same as \vec{a} if ฮป > 0, opposite if ฮป < 0
๐น Properties of Scalar Multiplication
- p(\vec{a} + \vec{b}) = p\vec{a} + p\vec{b}
- (p + q)\vec{a} = p\vec{a} + q\vec{a}
- p(q\vec{a}) = (pq)\vec{a}
๐ก Note: To prove \vec{a} is parallel to \vec{b}, show \vec{a} = ฮป\vec{b}.
๐ COMPONENTS OF A VECTOR
๐น Two Dimensions
\vec{OP} = x\hat{i} + y\hat{j}
|\vec{OP}| = √(x² + y²)
๐น Three Dimensions
\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}
|\vec{OP}| = √(x² + y² + z²)
๐น Vector Joining Two Points
If P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), then:
\vec{P₁P₂} = (x₂ - x₁)\hat{i} + (y₂ - y₁)\hat{j} + (z₂ - z₁)\hat{k}
๐ SECTION FORMULA
๐น Internal Division
Point R dividing AB in ratio m : n:
\vec{OR} = (m\vec{b} + n\vec{a})/(m + n)
๐น External Division
\vec{OR} = (m\vec{b} - n\vec{a})/(m - n)
๐น Midpoint
\vec{OR} = (\vec{a} + \vec{b})/2
● DOT PRODUCT (SCALAR PRODUCT)
Dot Product
\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos ฮธ
where ฮธ is the angle between \vec{a} and \vec{b}.
๐ก Note: \vec{a} \cdot \vec{b} is a real number (scalar).
๐น Properties of Dot Product
- \vec{a} \cdot \vec{a} = |\vec{a}|²
- \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1
- \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0
- \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} (Commutative)
- \vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}
๐น Angle Between Vectors
cos ฮธ = (\vec{a} \cdot \vec{b})/(|\vec{a}| |\vec{b}|)
✖ CROSS PRODUCT (VECTOR PRODUCT)
Cross Product
\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin ฮธ · \hat{n}
where \hat{n} is a unit vector perpendicular to both \vec{a} and \vec{b}.
๐ก Note: \vec{a} \times \vec{b} is a vector quantity.
Magnitude: |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin ฮธ
๐น Properties of Cross Product
- \vec{a} \times \vec{a} = \vec{0}
- \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0
- \hat{i} \times \hat{j} = \hat{k}
- \hat{j} \times \hat{k} = \hat{i}
- \hat{k} \times \hat{i} = \hat{j}
- \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} (Anti-commutative)
- \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}
๐น Area of Triangle
Area = (1/2)|\vec{AB} \times \vec{AC}|
๐ MCQ QUESTIONS
Q1. A vector quantity has:
(a) Only magnitude (b) Only direction (c) Both magnitude and direction (d) None
Answer: (c) Both magnitude and direction
Q2. The magnitude of zero vector is:
(a) 0 (b) 1 (c) -1 (d) Undefined
Answer: (a) 0
Q3. A unit vector in the direction of \vec{a} is:
(a) \vec{a}/|\vec{a}| (b) |\vec{a}|/\vec{a} (c) \vec{a} \times \vec{a} (d) \vec{a} \cdot \vec{a}
Answer: (a) \vec{a}/|\vec{a}|
Q4. If \vec{a} = ฮป\vec{b}, then \vec{a} and \vec{b} are:
(a) Equal (b) Collinear (c) Perpendicular (d) None
Answer: (b) Collinear
Q5. The dot product \vec{a} \cdot \vec{b} is:
(a) Vector (b) Scalar (c) Matrix (d) None
Answer: (b) Scalar
Q6. The cross product \vec{a} \times \vec{b} is:
(a) Vector (b) Scalar (c) Zero (d) None
Answer: (a) Vector
Q7. \vec{a} \times \vec{a} is equal to:
(a) \vec{a} (b) 1 (c) \vec{0} (d) |\vec{a}|
Answer: (c) \vec{0}
Q8. \hat{i} \cdot \hat{i} is equal to:
(a) 0 (b) 1 (c) -1 (d) \hat{i}
Answer: (b) 1
Q9. \hat{i} \times \hat{j} is equal to:
(a) \hat{i} (b) \hat{j} (c) \hat{k} (d) 0
Answer: (c) \hat{k}
Q10. The position vector of point (3, 4, 5) is:
(a) 3\hat{i} + 4\hat{j} + 5\hat{k} (b) 3\hat{i} + 4\hat{j} (c) 4\hat{j} + 5\hat{k} (d) None
Answer: (a) 3\hat{i} + 4\hat{j} + 5\hat{k}
Q11. The magnitude of vector 3\hat{i} + 4\hat{j} is:
(a) 7 (b) 5 (c) 12 (d) 1
Answer: (b) 5
Solution: |v| = √(9+16) = √25 = 5
Q12. The magnitude of vector 2\hat{i} + 3\hat{j} + 6\hat{k} is:
(a) 11 (b) 7 (c) 13 (d) 9
Answer: (b) 7
Solution: |v| = √(4+9+36) = √49 = 7
Q13. The dot product of two perpendicular vectors is:
(a) 0 (b) 1 (c) -1 (d) |\vec{a}||\vec{b}|
Answer: (a) 0
Q14. The cross product of two parallel vectors is:
(a) 0 (b) 1 (c) \vec{0} (d) |\vec{a}||\vec{b}|
Answer: (c) \vec{0}
Q15. The sum of \vec{a} + (-\vec{a}) is:
(a) \vec{a} (b) 2\vec{a} (c) \vec{0} (d) None
Answer: (c) \vec{0}
✅ ANSWER KEY - MCQs
| Q.No. | Answer | Q.No. | Answer |
|---|---|---|---|
| 1 | (c) | 9 | (c) |
| 2 | (a) | 10 | (a) |
| 3 | (a) | 11 | (b) |
| 4 | (b) | 12 | (b) |
| 5 | (b) | 13 | (a) |
| 6 | (a) | 14 | (c) |
| 7 | (c) | 15 | (c) |
| 8 | (b) |
๐ KEY FORMULAS SUMMARY
| Concept | Formula |
|---|---|
| Magnitude | |v| = √(x² + y² + z²) |
| Unit Vector | รข = a/|a| |
| Dot Product | a·b = |a||b|cos ฮธ |
| Cross Product | a×b = |a||b|sin ฮธ·n̂ |
| Internal Division | OR = (mb + na)/(m+n) |
| External Division | OR = (mb - na)/(m-n) |
| Midpoint | OR = (a + b)/2 |
| Direction Cosines | l² + m² + n² = 1 |
| Area of Triangle | 1/2|AB × AC| |
๐น Quick Reference:
- i·i = j·j = k·k = 1
- i·j = j·k = k·i = 0
- i×j = k, j×k = i, k×i = j
- j×i = -k, k×j = -i, i×k = -j
๐ CBSE | Class XII Mathematics
Chapter 10: Vector Algebra | Study Notes