CLASS XII MATHEMATICS CHAPTER 10

Vector Algebra

Complete Class XII Mathematics chapter notes covering vectors, magnitude, unit vectors, direction cosines, vector operations, section formula, scalar product, vector product, scalar triple product and important applications.

CBSE Pattern Vector Representation Dot Product Cross Product Triple Product HOT Questions

Start Learning Enroll Now

Chapter Overview

Introduction

Vector Algebra provides an algebraic method for representing quantities having both magnitude and direction. Vectors are extensively used in geometry, mechanics, physics and three-dimensional mathematics.

In this chapter we study vector representation, magnitude, unit vectors, direction cosines, vector addition, scalar multiplication, section formula, scalar product, vector product and scalar triple product.

Magnitude

Measures the length of a vector.

Direction

Describes the orientation of a vector.

Operations

Vectors can be added and multiplied in meaningful ways.

Fundamentals

Basic Concepts of Vectors

Scalar and Vector Quantities

A scalar quantity is completely described by its magnitude, whereas a vector quantity requires both magnitude and direction.

Scalar

• Mass
• Time
• Temperature
• Distance

Vector

• Displacement
• Velocity
• Acceleration
• Force

\[ \vec a = a_1\hat i+a_2\hat j+a_3\hat k \]

Here \(a_1,a_2,a_3\) are the rectangular components of vector \(\vec a\).

Length of a Vector

Magnitude and Unit Vector

Magnitude

\[ |\vec a| = \sqrt{a_1^2+a_2^2+a_3^2} \]

Unit Vector

A vector having magnitude \(1\) and the same direction as \(\vec a\) is called the unit vector in the direction of \(\vec a\).

\[ \hat a=\frac{\vec a}{|\vec a|} \]

Important: A non-zero vector is required to define its unit vector.

Direction

Direction Cosines

If a vector makes angles \(\alpha,\beta,\gamma\) with the positive \(x,y,z\)-axes respectively, then

\[ l=\cos\alpha,\qquad m=\cos\beta,\qquad n=\cos\gamma \]

\[ l^2+m^2+n^2=1 \]

Vector Operations

Vector Addition

If two vectors are represented by \(\vec a\) and \(\vec b\), their sum is obtained component-wise.

\[ \vec a+\vec b = (a_1+b_1)\hat i +(a_2+b_2)\hat j +(a_3+b_3)\hat k \]

Properties

\[ \vec a+\vec b=\vec b+\vec a \]

Commutative property

\[ (\vec a+\vec b)+\vec c = \vec a+(\vec b+\vec c) \]

Associative property

Division of a Line Segment

Section Formula

Let \(P\) divide the line segment joining the points represented by position vectors \(\vec a\) and \(\vec b\) internally in the ratio \(m:n\).

\[ \boxed{ \vec p= \frac{m\vec b+n\vec a}{m+n} } \]

Midpoint

\[ \vec p=\frac{\vec a+\vec b}{2} \]

External Division

\[ \boxed{ \vec p= \frac{m\vec b-n\vec a}{m-n} } \]

Scalar Product

Dot Product

The scalar product of two vectors \(\vec a\) and \(\vec b\) is defined as

\[ \boxed{ \vec a\cdot\vec b = |\vec a||\vec b|\cos\theta } \]

Component Form

\[ \vec a\cdot\vec b = a_1b_1+a_2b_2+a_3b_3 \]

Perpendicular

\[ \vec a\cdot\vec b=0 \]

Acute Angle

\[ \vec a\cdot\vec b>0 \]

Obtuse Angle

\[ \vec a\cdot\vec b<0 \]

Vector Product

Vector Product (Cross Product)

The vector (cross) product of two vectors \(ec a\) and \(ec b\) is defined as \[ ec a imes ec b = |ec a| |ec b| \sin heta \, \hat n \] where \( heta\) is the angle between \(ec a\) and \(ec b\) (\(0 \le heta \le \pi\)), and \(\hat n\) is a unit vector perpendicular to both \(ec a\) and \(ec b\) such that \(ec a, ec b, \hat n\) form a right-handed system.

Component Form (Determinant)

\[ ec a imes ec b = \begin{vmatrix} \hat i & \hat j & \hat k \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 nd{vmatrix} \]

Area of Parallelogram & Triangle

\( ext{Area of parallelogram with adjacent sides } ec a, ec b = |ec a imes ec b|\)
\( ext{Area of triangle with adjacent sides } ec a, ec b = rac{1}{2}|ec a imes ec b|\)