Vector Algebra Class 12th Notes - Free NCERT Class 12 Maths Chapter 10 Notes

Vector Algebra Class 12th Notes - Free NCERT Class 12 Maths Chapter 10 Notes - Download PDF

Edited By Ramraj Saini | Updated on Apr 23, 2022 01:09 PM IST

Class 12 Math chapter 10 notes are regarding Vector Algebra. In chapter 10 we will be going through the vector concepts in Vector Algebra Class 12 notes. This Class 12 Maths chapter 10 notes contains the following topics: vectors, representation, position vector, magnitude, types of vectors, addition vectors, direction cosines, addition vectors, properties.

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NCERT Class 12 Math chapter 10 notes also contain standard formulas that are to be remembered for the implementation in problems. NCERT Class 12 Math chapter 10 contains a detailed explanation of topics, examples, exercises. By going through the document students can cover all the topics that are in NCERT Notes for Class 12 Math chapter 10 textbook. It also contains examples, exercises, a few interesting points and most importantly contains FAQ’s that are frequently asked questions by students which can clarify many other students with the same doubt. Every concept that is in CBSE Class 12 Maths chapter 10 notes is explained here in a simple and understanding way that can reach students easily. The notes for Class 12 Maths chapter 10 will help get detailed information of vector algebra.

All these concepts can be downloaded from Class 12 Maths chapter 10 notes pdf download, Class 12 Vector Algebra notes, Vector Algebra Class 12 notes pdf download.

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NCERT Class 12 Chapter 10 Notes

Vectors: Quantities that have both magnitude and direction, and follow vector addition laws, are called vectors.

Denoted by: $\vec{AB}$

Scalars: Quantities that have only magnitude but no direction are called scalars.

Example for Vectors and Scalars:

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The magnitude of the vector: This shows the value of the vector and is denoted by: $|\vec{AB}|$

Position Vector: Let O(0,0,0) be the origin and P(x,y,z) be the position vector.

Then the magnitude of a vector is denoted by:

Formula:

Direction cosines: α, β and γ are the angles along the direction of the position vector along with the axis OX, OY, OZ. cos α, cos β, cos γ are the three positional vectors.

$\\ cos\alpha = cos\beta = cos \gamma \\ \\ \frac{l}{a} = \frac{m}{b} = \frac{n}{c} \\ \\ l^2+m^2+n^2 = 1$

Pictorial Representation of Direction cosines:

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Types of vectors:

Zero vector: A vector whose magnitude is zero is called a Zero vector. It is denoted by
Unit Vector: A vector whose magnitude is equal to One is called a Unit vector.
Coinitial Vectors: Two or more vectors are called coinitial if they have the same initial point.
Collinear Vectors: If two vectors are parallel to a given line then the vectors are called collinear.
Equal Vectors: The vectors that have the same magnitude and equal direction such vectors are called Equal vectors ${\left |\vec{a}\right |}={\left |\vec{b}\right |}$
Negative of a Vector: Vectors whose magnitude is the same but opposite in direction are called negative vectors
Addition of vectors:

Triangle law of addition:

If two vectors are represented in the same direction and the resultant is represented in the opposite direction that is called the Triangle law of addition.

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Parallelogram law of vectors addition:

If two vectors are adjacent sides of the parallelogram then the resultant vector is the diagonal of sides. ${\left |\vec{OA}\right |}+{\left |\vec{OC}\right |}={\left |\vec{OB}\right |}$

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Properties of vector addition:

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Multiplication of a Vector by a Scalar:

$\text {where}\ \vec{a} \ \text{denotes a vector}\ \lambda \ \text{denotes a scalar.}$

The magnitude of

Properties of Multiplication of vectors:

$\\a)\alpha \left ( \vec{a}+\vec{b} \right )=\alpha \vec{a}+\alpha \vec{b} \\b)\left (\alpha +\beta \right )\vec{a}=\alpha\vec{a}+\beta\vec{a} \\c)\alpha \left ( \beta \vec{a} \right )=\left ( \alpha \beta \right )\vec{a}$

Component of Vectors:

Let the position vector be is a component of the vector.

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Two Dimension:

$\\ \vec{OP} =x\hat{i}+y\hat{j} \\ \left | \vec{OP} \right |=\sqrt{x^{2}+y^{2}}$

Three Dimension:

$\\\left | \vec{OP} \right |=x\hat{i}+y\hat{j}+z\hat{k} \\\left | \vec{OP} \right |=\sqrt{x^{2}+y^{2}+z^{2}}$

Vectors joining two Points: are any points on the axis.

Vector joining is given below:

$\\ \vec{P_{1}P_{2}} =(x_{2}-x_{1})\hat{i}+ (y_{2}-y_{1})\hat{j}+(z_{2}-z_{1})\hat{k} \\\left | \vec{P_{1}P_{2}} \right |=\sqrt{(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$

Section Formula: $\vec{OP}$ be a position vector and line segment AB

Divided internally in m:n ratio, then the formula is $\ \vec{OP}=\frac {m\vec{b}+n\vec{a}}{m+n}$
Divided externally in the m:n ratio, the formula is $\vec{OP}=\frac {m\vec{b}-n\vec{a}}{m-n}$

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Midpoints of vectors: $\vec{OP}=\frac {\vec{a}+\vec{b}}{2}$

Dot Product of Vectors: θ is used to find the angle between the vectors $\vec{a} \ and \ \vec{b}$

Scalar/dot product is denoted by: $\vec{a}\cdot\vec{b}$

Formula: $\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta$

Properties of dot product:

$\\\vec{a}\cdot\vec{a}=\left |\vec{a} \right |^{2}\Rightarrow \\\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1$

$\\\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$

$\\\vec{a}\cdot\vec{0}=0$

$\\\vec{a}\cdot\vec{b}=\left ( acos\theta \right )b=\left ( \text{projection of}\;\vec{a}\;\text{on}\;\vec{b} \right )b=\left ( \text{projection of}\;\vec{b}\;\text{on}\;\vec{a} \right )a$

$\\\vec{a}\left ( \vec{b}+\vec{c} \right )=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}\;\left (\text{Distributive Law} \right )$

$\\\left ( l\vec{a} \right )\cdot\left ( m\vec{b} \right )=lm\left ( \vec{a}\cdot\vec{b} \right )$

$\\\vec{a}\cdot\vec{b}=0\Leftrightarrow \vec{a}\cdot\vec{b}\;\text{are perpendicular to each other}\Leftrightarrow \hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0$

$\\\left (\vec{a}\pm \vec{b} \right )^{2}=\left (\vec{a}\pm \vec{b} \right )\cdot\left (\vec{a}\pm \vec{b} \right )=\vec{a}^{2}+\vec{b}^{2}\pm 2\vec{a}\cdot\vec{b}$

$\\\left (\vec{a}+\vec{b} \right )\cdot\left (\vec{a}-\vec{b} \right )=\vec{a}^{2}-\vec{b}^{2}=a^{2}-b^{2}$

$\\If\ \vec{a}=a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k} \ and\ \vec{b}=b_{1}\hat{i}+b_{2}\hat{j}+b_{3}\hat{k}\\ then \ \vec{a}\cdot\vec{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}.$

$\\If\ \vec{a} \ ,\ \vec{b}\ \text{are non-zero , angle between them can be given by} \\\cos\theta=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\cdot|\vec{b}|} =\frac{a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}\cdot\sqrt{b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}}$

Projection of vector along another vector:

$\\ Projection\ of\ \vec{a} \ on \ \vec{b}\ is\ given\ by:\ \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \\ Projection\ of\ \vec{b}\ on\ \vec{a}\ is\ given\ by:\ \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}$

$\\ If \ \theta = 0\ \text{projection of}\ \vec{AB} \ will \ be\ \vec{AB}\ only.\\ \\ If\ \theta = \pi \ \text{projection of}\ \vec{AB}\ will\ be\ \vec{BA}. \\ \\ If\ \theta = \frac{\pi}{2} \ or \ \frac{3 \pi}{2}\ projection\ of\ \vec{AB}\ will\ be\ zero$

Cross Product:

$\\ \theta\ \text{is the angle between two non-parallel vectors}\ \vec{a}\ and\ \vec{b}\ ,\\ \text{then the cross product is given by}\ \vec{a}\times\vec{b}.\\ Formula: \vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta$

Property of Cross Product:

The cross product is defined as

$\\ \vec{a}\times\vec{b}=\left (|\vec{a}||\vec{b}|\sin\theta \right )\hat{n} \\where \ \hat{n}\ \text{is the unit vector orthogonal to both}\ \hat{a}\ and\ \hat{b} \\ \text{following the right-hand rule for direction, and } \theta \text{ is angle between the vectors}$

$\\If \ \vec{a}=[a_{1},a_{2},a_{3}]\ and\ \vec{b}=[b_{1},b_{2},b_{3}] \ then, \\\vec{a}\times\vec{b}=[a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1}]$

The magnitude of the cross product vector is

$\\|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin\theta,\\ \text{which also calculates the area of the parallelogram defined by } \vec{a} \ and\ \vec{b}$

$\\\text{ The angle}\ \theta\text{ , between two vectors, }\vec{a} \ and\ \vec{b} \text{ is given by}\\\theta=\sin^{-1}\left (\frac{|\vec{a}|\times|\vec{b}|}{|\vec{a}|\cdot|\vec{b}|} \right )$

$\text{ For any vectors } \vec{a},\vec{v} \ and\ \vec{w} \text{ and scalar k}\ \in \mathbb{R}$

$\\\vec{u}\times\vec{v}=-\left ( \vec{v}\times\vec{u} \right ) \\\vec{u}\times\left ( \vec{v}+\vec{w} \right )=\vec{u}\times\vec{v}+\vec{u}\times\vec{w} \\\left ( \vec{u}+\vec{v} \right )\times\vec{w}=\vec{u}\times\vec{w}+\vec{v}\times\vec{w}$

$\\If \ \vec{u} \ and\ \vec{v} \ are \ non\ zero,\ \vec{u}\times\vec{v}=0 \\\text{ if and only there is a scalar m} \in \mathbb{R} \ such \ that\ \vec{u}=m\vec{v}$

$k\left ( \vec{u}\times\vec{v} \right )=\left ( k\vec{u} \right )\times\vec{v}=\vec{u}\times\left ( k\vec{v} \right )$

With this topic we conclude NCERT class 12 chapter 10 notes.

The link for the NCERT textbook pdf is given below:

URL: ncert.nic.in/textbook/pdf/lemh204.pdf

Significance of NCERT Class 12 Maths Chapter 10 Notes:

NCERT Class 12 Maths chapter 10 notes will be very much helpful for students to score maximum marks in their 12 board exams. In Vector Algebra Class 12 chapter 10 notes we have discussed many topics: vectors, scalars, representation, position vector, magnitude, types of vectors, addition vectors, direction cosines, addition vectors, dot product, cross product, projections, with pictorial representations and their properties. NCERT Class 12 Mathematics chapter 10 is also very useful to cover major topics of Class 12 CBSE Mathematics Syllabus.

The CBSE Class 12 Maths chapter 10 will help to understand the formulas, statements, rules with their conditions in detail. This pdf also contains previous year’s questions and NCERT textbook pdf. The next part contains FAQs most frequently asked questions along with topic-wise explanations. By referring to the document you can get a complete idea of all the topics of Class 12 chapter 10 Vector product pdf download.

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