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

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.

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.

Also, students can refer,

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:


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: 1646303701751

Formula: 1646303701505

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:


Types of vectors:

  • Zero vector: A vector whose magnitude is zero is called a Zero vector. It is denoted by1646304564383

  • Unit Vector: A vector whose magnitude is equal to One is called a Unit vector. 1646304563869

  • 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 1646304657754
  • 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.


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 |}


Properties of vector addition:


Multiplication of a Vector by a Scalar:

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

The magnitude of 1646305593837

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 ayRcn4nGVpeIMFiUvfqnYiRIoTOktegQv6TvTWhjKSQiHMh16LYy6Inbc217CyNMaW2vRIKJp8NFjVkDu03-EaOomHpW6sLq0_fbkvyZ-86Z11B45HlQ2QNRiDyHbQis a component of the vector.


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:1646308617622 are any points on the axis.

Vector joining1646308617958 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}=\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:


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.

NCERT Class 12 Notes Chapter Wise.

Subject Wise NCERT Exemplar Solutions

Subject Wise NCERT Solutions

NCERT Books and Syllabus

Frequently Asked Question (FAQs)

1. Why Vector Algebra Class 12 notes are important?

Vector Algebra Class 12 notes are used to find different relations among vector quantities.

2. Discuss important laws of vectors in Class 12 Math chapter 10 notes?

Commutative, Associative, distributive laws are important in vector algebra.

3. Who were the founders of NCERT Class 12 Math chapter 10 notes?

Josiah Willard Gibbs and Oliver Heaviside were the inventors of vector algebra.

4. What will the angle be if the vectors are negative according to NCERT notes for Class 12 Maths chapter 10?

Angle between negative vectors will be 180 degrees.

5. Is dot product associative?

According to Class 12 Vector Algebra notes, the dot product is not associative because we need 3 vectors for associative law but we only have 2 vectors in the dot product, and this is possible in vector products.


Get answers from students and experts

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg


An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)


Option 2)

\; K\;

Option 3)


Option 4)


In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)


Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)


Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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