CBSE Class 12 Mathematics Notes Chapter 10 Vector Algebra

Vector is a quantity having both magnitude and direction, such as displacement, velocity, force, and acceleration. Graphically, a vector is represented by a directed line segment.

• It is denoted by a letter with an arrow over it, as in a, or in bold type, as in \(\overrightarrow{a}\).
• A vector whose initial and terminal points are the same is called a zero vector.
• The position vector r or any point P with respect to the origin of reference O is a vector OP.
• A point with position vector r will be written as P(r).

The important points that are covered in this class 12 Maths vector algebra chapter include vector introduction, types of vectors, and operations on vectors like addition. CBSE Class 12 Mathematics Notes for Chapter 8 Application of Integrals are given in the article below for easy preparation and understanding of the concepts involved.

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Scalar and Vector Quantities

• Scalar Quantities: The quantities that have only magnitude and no direction, and do not obey the rules of vector algebra, are called scalar quantities.
• Vector Quantities: The quantities that have both magnitude and direction and obey the rules of vector algebra, are called vector quantities.

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Representation of a Vector

• A vector is generally represented by a direction line segment, say \(\overrightarrow{AB}\).
• A is called the initial point and B is called the terminal point.
• The modulus of the vector \(\overrightarrow{AB}\) is denoted as |\(\overrightarrow{AB}\)|.

Vector representation

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Types of Vectors

• Zero Vector: A vector of zero magnitude i.e. which has the same initial and terminal points, is called a zero vector.
• Unit Vector: A vector of unit magnitude in the direction of a vector a is called \(\overrightarrow{a}\) unit vector and is denoted by \(\hat{a}\).
• Equal Vector: If the magnitude and direction of two vectors are the same, then those are called equal vectors.
• Like Vectors: Two parallel vectors are said to be like when they have the same sense of direction i.e. angle between them is zero.
• Unlike Vectors: Two parallel vectors are said to be unlike vectors when they have the opposite sense of direction i.e. angle between them is π.
• Collinear Vectors: Two or more vectors are known as collinear vectors if they are parallel to a given straight line. The magnitude of collinear vectors can be different.
• Coplanar Vectors: Vectors are said to be coplanar if they occur in the same or common plane.

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Addition of Vectors

• Triangle Law: If two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) lie along the two sides of a triangle in consecutive order, then the third side represents the sum \(\overrightarrow{a}\) + \(\overrightarrow{b}\).

Triangle Law

• Parallelogram Law: If two vectors lie along two adjacent sides of a parallelogram then the diagonal of the parallelogram through the common vertex represents their sum.

Parallelogram Law

• Polygon Law: If (n - 1) sides of a polygon represent vectors \(\vec{a1}, \vec{a2}, \vec{a3}\), ...., in consecutive order, then n^th represents their sum.

Polygon Law

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Properties of Vector Addition

• Vector addition is commutative, i.e.

\(\vec{a} + \vec{b} = \vec{b} + \vec{a}\)

• Vector addition is associative, i.e.

\((\vec{a} + \vec{b} )+ \vec{c} = \vec{a} +( \vec{b} + \vec{c} )\)

• \(\vec{a} + \vec{O} = \vec{a} = \vec{O} + \vec{a}\), where \(\vec{O}\) is the additive identity.
• \(\vec{a} + \vec{(-a)} = \vec{O} = \vec{(-a)} + \vec{a}\), where \(\vec{(-a)}\) is additive inverse.

Properties of Multiplication of Vector by a Scalar

• If A and B are two scalars, then
  • \(A \vec{(Ba)} = AB\vec{a}\)
  • \((A+B)\vec{a} = A\vec{a}+B\vec{a}\)
  • \(A(\vec{a} + \vec{b}) = A\vec{a} +A\vec{b}\)

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Components of Vector in Two Dimensions

• The process of splitting a vector is called the resolution of a vector.
• If A_x and A_y are the magnitudes of Axand Ay then A_x\(\hat{i}\) and A_y\(\hat{j}\) are the vector components of A in the x and y directions respectively.

\(\vec{A}\) = \(\vec{A}\)_x + \(\vec{A}\)_y = \(\vec{A}\)_x\(\hat{i}\) + \(\vec{A}\)_y\(\hat{j}\)

And, |\(\vec{A}\)| = \(\sqrt{A_x^2 + A_y^2}\)

Components of a Vector in two Dimensions

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Components of a Vector in Three Dimensions

• In three dimensions we can define a unit vector in the x-direction by \(\hat{i}\) or \(\hat{x}\), in the y-direction by \(\hat{j}\) or, \(\hat{y}\) and in the z-direction by \(\hat{k}\) or \(\hat{z}\).
• The vector A can be represented by

\(\vec{A}\) = \(\vec{A}\)_x + \(\vec{A}\)_y + \(\vec{A}\)_z = \(\vec{A}\)_x\(\hat{i}\) + \(\vec{A}\)_y\(\hat{j}\) + \(\vec{A}\)_z\(\hat{z}\)

And |\(\vec{A}\)| = \(\sqrt{A_x^2+A_y^2+A_z^2}\)

Components of a Vector in Three Dimensions

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Scalar Product or Dot Product

• Let \(\vec{a} \) and \(\vec{b} \) be two non-zero vectors inclined at an angle θ.
• Then the scalar product of \(\vec{a} \) and \(\vec{b} \) is denoted by \(\vec{a} \) .\(\vec{b} \) and defined as

\(\vec{a} \) . \(\vec{b} \) = ab cos θ

Scalar Product

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Vector Product or Cross Product

• The vector product of two vectors \(\vec{a}\) and \(\vec{b}\) is a vector.
• It is represented by

\(\vec{a}\) x \(\vec{b}\) = ab sin θ

Vector Product

There are Some important List Of Top Mathematics Questions On Vector Algebra Asked In CBSE CLASS XII