Class 12 Mathematics (041) · Chapter 10 NotesVector Algebra
Master Class 12 Vector Algebra with NCERT-based notes. Learn about dot product, cross product, direction cosines, and vector components for CBSE exams.
Vector algebra is a fundamental branch of mathematics that bridges the gap between geometry and algebra, providing essential tools for describing physical quantities in three-dimensional space. While scalar quantities like mass, time, and temperature are defined solely by their magnitude, vector quantities such as force, velocity, and displacement require both magnitude and direction for a complete description. This chapter introduces students to the mathematical representation of directed line segments and the algebraic operations that govern them. Students will learn to represent vectors in component form using unit vectors along the coordinate axes, calculate magnitudes, and determine direction cosines. The curriculum covers critical operations including vector addition, multiplication by scalars, and the two primary types of vector products: the scalar (dot) product and the vector (cross) product. Mastering these concepts is vital for solving complex problems in physics and engineering, as they allow for the precise calculation of angles, projections, and the areas of geometric figures like triangles and parallelograms.
What you'll learn
1Distinguish between scalar and vector quantities with real-world examples.
2Represent vectors geometrically as directed line segments and algebraically in component form.
3Calculate the magnitude, direction cosines, and direction ratios of a given vector.
4Perform vector addition and subtraction using the triangle and parallelogram laws.
5Apply the section formula to find the position vector of a point dividing a line segment.
6Compute the scalar (dot) product to find the angle between vectors and their projections.
7Utilize the vector (cross) product to determine perpendicular vectors and calculate geometric areas.
Chapter at a glance
01Vectors and Scalars
02Addition and Subtraction of Vectors
03Components of a Vector and Unit Vector
04Dot Product and Cross Product
05Scalar Triple Product and Vector Triple Product
06Applications of Vector Algebra
Detailed chapter notes
01
Basic Concepts and Types of Vectors
A vector is defined as a quantity that possesses both magnitude and direction, represented geometrically by a directed line segment. The starting point is the initial point, and the ending point is the terminal point. The magnitude is the length of this segment. Several specific types of vectors form the foundation of this study. A zero vector has coinciding initial and terminal points, resulting in zero magnitude and an undefined direction. A unit vector has a magnitude of exactly one unit. Coinitial vectors share the same starting point, while collinear vectors are parallel to the same line regardless of their magnitude. Equal vectors must have both the same magnitude and direction, even if their positions in space differ. A negative vector has the same magnitude as a given vector but points in the exactly opposite direction.
Magnitude of vector AB is denoted as |AB|.
Unit vector in the direction of 'a' is denoted as a-hat.
Free vectors can be shifted parallel to themselves without changing their value.
02
Position Vectors and Direction Cosines
In a three-dimensional rectangular coordinate system, the position vector of a point P(x, y, z) is the vector starting from the origin O(0, 0, 0) and ending at P. Its magnitude is calculated using the distance formula as the square root of (x² + y² + z²). The direction of this vector is described by direction angles (alpha, beta, and gamma) which the vector makes with the positive x, y, and z axes. The cosines of these angles are known as direction cosines (l, m, n). An important property is that the sum of the squares of the direction cosines always equals one (l² + m² + n² = 1). Direction ratios are any numbers proportional to the direction cosines, typically represented by the scalar components of the vector.
Position vector r = xi + yj + zk.
l = cos(alpha), m = cos(beta), n = cos(gamma).
Direction ratios (a, b, c) are related to cosines by l = a/r, m = b/r, n = c/r.
03
Vector Addition and Scalar Multiplication
Vector addition follows specific geometric rules. The triangle law states that if two vectors are represented by two sides of a triangle taken in order, their sum is the third side taken in the opposite direction. Alternatively, the parallelogram law states that if two coinitial vectors form the adjacent sides of a parallelogram, their sum is the diagonal passing through their common initial point. Vector addition is both commutative and associative. Multiplication of a vector by a scalar 'k' results in a new vector whose magnitude is |k| times the original. If 'k' is positive, the direction remains the same; if 'k' is negative, the direction is reversed. This operation is distributive over vector addition.
Commutative Lawa + b = b + a.
Associative Law(a + b) + c = a + (b + c).
Additive Identitya + 0 = a.
04
Components of a Vector and Section Formula
Vectors are most easily managed using their component form, utilizing unit vectors i, j, and k along the x, y, and z axes respectively. Any vector can be expressed as r = xi + yj + zk, where x, y, and z are scalar components. When joining two points P1(x1, y1, z1) and P2(x2, y2, z2), the vector is found by subtracting the coordinates of the initial point from the terminal point. The section formula is used to find the position vector of a point R that divides the line segment joining P and Q in a given ratio m:n. For internal division, the formula is (mb + na) / (m + n), and for external division, it is (mb - na) / (m - n).
Magnitude |r| = √(x² + y² + z²).
Vector joining P1 and P2(x2-x1)i + (y2-y1)j + (z2-z1)k.
Midpoint formula(a + b) / 2.
05
Scalar or Dot Product
The scalar product of two vectors 'a' and 'b' is a real number defined as the product of their magnitudes and the cosine of the angle between them (a . b = |a||b|cosθ). If the dot product is zero, the two non-zero vectors are perpendicular. The dot product is commutative and distributive over addition. In component form, it is the sum of the products of corresponding components: a1b1 + a2b2 + a3b3. A significant application of the dot product is finding the projection of one vector onto another. The projection of vector 'a' on vector 'b' is given by (a . b) / |b|. This represents the scalar length of the 'shadow' of vector 'a' along the direction of 'b'.
a . b = 0 implies a is perpendicular to b.
i . i = j . j = k . k = 1.
i . j = j . k = k . i = 0.
06
Vector or Cross Product
The vector product of two vectors 'a' and 'b' results in a new vector perpendicular to both. It is defined as a x b = |a||b|sinθ n-hat, where n-hat is a unit vector perpendicular to the plane of 'a' and 'b' following the right-hand thumb rule. If the cross product is a zero vector, the vectors are parallel. Unlike the dot product, the cross product is not commutative; reversing the order changes the sign (a x b = -b x a). The magnitude of the cross product has geometric significance: it equals the area of a parallelogram with 'a' and 'b' as adjacent sides. Half of this magnitude gives the area of the triangle formed by the same vectors.
a x b = 0 implies a is parallel to b.
i x j = k, j x k = i, k x i = j.
Area of parallelogram = |a x b|.
Want the complete chapter resources?Topic notes, quizzes and flashcards for Vector Algebra.
Temperature has only magnitude and no direction, making it a scalar quantity. Displacement, velocity, and force all require direction specification and are vectors.
Question 02
If vector **a** = 2**i** + 3**j** and vector **b** = 4**i** + 5**j**, then **a** + **b** equals:
A6**i** + 8**j**
B2**i** + 2**j**
C-2**i** - 2**j**
D8**i** + 15**j**
Show answer
Answer: (A) 6**i** + 8**j**
Vector addition is performed component-wise: (2+4)**i** + (3+5)**j** = 6**i** + 8**j**.
Question 03
If a vector has components (3, 4) in 2D, what is its magnitude?
A5
B7
C√7
D12
Show answer
Answer: (A) 5
Magnitude = √(3² + 4²) = √(9 + 16) = √25 = 5. This is the direct application of the distance formula for vector magnitude.
Question 04
If vectors **a** = 2**i** + 3**j** + **k** and **b** = **i** + **j** + **k**, then **a** · **b** equals:
A5
B6
C7
D8
Show answer
Answer: (B) 6
Dot product: **a** · **b** = (2)(1) + (3)(1) + (1)(1) = 2 + 3 + 1 = 6. This is direct application of the dot product formula.
Question 05
If **a**, **b**, and **c** are three vectors, then the scalar triple product **a** · (**b** × **c**) is also known as:
ABox product or mixed product
BCross product
CDot product
DScalar product
Show answer
Answer: (A) Box product or mixed product
The scalar triple product **a** · (**b** × **c**) represents the box product or mixed product of three vectors, which gives a scalar value representing the volume of the parallelepiped formed by the three vectors.
Ready for more practice?Unlock the full quiz for this chapter.
What is the difference between a scalar and a vector quantity?
A scalar quantity, such as mass or temperature, is described by a single real number representing magnitude. A vector quantity, such as force or velocity, requires both a magnitude and a specific direction to be fully defined.
How do you determine if two vectors are perpendicular using algebra?
Two non-zero vectors are perpendicular if their scalar (dot) product is equal to zero. This is because the cosine of 90 degrees is zero, making the entire expression |a||b|cos(90°) vanish.
What are direction cosines and how are they related to direction ratios?
Direction cosines are the cosines of the angles a vector makes with the coordinate axes. Direction ratios are any set of three numbers that are proportional to these direction cosines, often represented by the vector's components.
Why is the cross product not commutative?
The cross product follows the right-hand rule to determine the direction of the resulting vector. When the order of vectors is swapped, the direction of the resultant vector is reversed, meaning a x b = -(b x a).
How can vector algebra be used to find the area of a triangle?
If two sides of a triangle are represented by vectors 'a' and 'b' originating from the same vertex, the area of the triangle is exactly half the magnitude of their cross product, calculated as 1/2 |a x b|.