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

Have you ever thought about how a pilot flies a plane in the right direction and speed? Or during sports commentary, how they calculate the trajectory of a ball. These are all applications of vector algebra, and in maths chapter 10 class 12, we read about this chapter. A vector is a quantity which has magnitude as well as direction. In Vector Algebra class 12 notes, students will learn about vectors, representation, position vector, magnitude, types of vectors, addition and multiplication of vectors, component of vectors, direction cosines, and other properties.

After completing the solutions from the textbook, students need a proficient study material for quick revision. That is when vector algebra class 12 NCERT notes come into play. These notes are prepared by Careers360 experts and cover all the important topics, formulas, and examples thoroughly. These Class 12 Maths Chapter 10 notes follow the latest CBSE guidelines and syllabus. Students can also check NCERT Exemplar Class 12 Maths Chapter 10 Solutions Vector Algebra for more exposure to this chapter.

NCERT Class 12 Chapter 10 Notes

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

Denoted by: $\overrightarrow{A B}$

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: $|\overrightarrow{A B}|$

Representation of a Vector

A vector is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector.

Position Vector: Let $\mathrm{O}(0,0,0)$ be the origin and $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ be the position vector.
Then the magnitude of a vector is denoted by: $|\overrightarrow{O P}|$
Formula: $|\overrightarrow{O P}|=\sqrt{x^2+y^2+z^2}$

Direction Cosines

Let $r$ be the position vector of a point $P(x, y, z)$. Then, the direction cosines of vector $r$ are the cosines of angles $\alpha, \beta$, and $y$ (i.e. $\cos \alpha$, $\cos \beta$, and $\cos \gamma$ ) that the vector $r$ makes with the positive direction of $X, Y$, and $Z$-axes respectively. Direction cosines are usually denoted by $I, m$, and $n$, respectively.

From the figure, note that ΔOAP is a right-angled triangle, and thus, we have

$
\cos \alpha=\frac{x}{r}(r \text { stands for }|r|)
$

Similarly, from the right angled triangles $O B P$ and $O C P$, We have,
$
\cos \beta=\frac{y}{r} \text { and } \cos \gamma=\frac{z}{r}
$

So we have the following results,
$
\begin{aligned}
& \cos \alpha=l=\frac{x}{\sqrt{x^2+y^2+z^2}}=\frac{x}{|\mathbf{r}|}=\frac{x}{r} \\
& \cos \beta=m=\frac{y}{\sqrt{x^2+y^2+z^2}}=\frac{y}{|\mathbf{r}|}=\frac{y}{r} \\
& \cos \gamma=n=\frac{z}{\sqrt{x^2+y^2+z^2}}=\frac{z}{|\mathbf{r}|}=\frac{z}{r}
\end{aligned}
$

Also,
$
\begin{aligned}
& l^2=\frac{x^2}{x^2+y^2+z^2} \\
& m^2=\frac{y^2}{x^2+y^2+z^2} \\
& n^2=\frac{z^2}{x^2+y^2+z^2}
\end{aligned}
$

Add (i), (ii) and (iii)
$
\begin{aligned}
& l^2+m^2+n^2=\frac{x^2}{x^2+y^2+z^2}+\frac{y^2}{x^2+y^2+z^2}+\frac{z^2}{x^2+y^2+z^2} \\
& \Rightarrow l^2+m^2+n^2=1 \\
& \Rightarrow \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1
\end{aligned}
$

The coordinates of the point P may also be expressed as (lr, mr, nr).

Types of vectors:

• Zero vector: A vector whose magnitude is zero is called a Zero vector. It is denoted by $|\overrightarrow{0}|$.
• Unit Vector: A vector whose magnitude is equal to one is called a Unit vector.
  $a=\frac{\vec{a}}{|\vec{a}|}$
• 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.
  $|\vec{a}|=|\vec{b}|$
• Negative of a Vector: Vectors whose magnitude is the same but opposite in direction are called negative vectors.
  $|\overrightarrow{A B}|=|-\overrightarrow{B A}|$

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.
$|\overrightarrow{O A}|+|\overrightarrow{O C}|=|\overrightarrow{O B}|$

Properties of vector addition:
Commutative: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$
Associative: $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$
Additive identity: $\vec{a}+\overrightarrow{0}=\vec{a}$
Additive inverse: $\vec{a}+(-\vec{a})=\overrightarrow{0}$

Multiplication of a Vector by a Scalar:
Where $\vec{a}$ denotes a vector and $\lambda$ denotes a scalar.
The magnitude of $|\lambda \cdot \vec{a}|=|\lambda| \cdot|\vec{a}|$

Properties of Multiplication of Vectors:
(a) $\alpha(\vec{a}+\vec{b})=\alpha \vec{a}+\alpha \vec{b}$
(b) $(\alpha+\beta) \vec{a}=\alpha \vec{a}+\beta \vec{a}$
(c) $\alpha(\beta \vec{a})=(\alpha \beta) \vec{a}$

Components of Vectors:
Let the position vector be $|\overrightarrow{O P}|=\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$ is a component of the vector.

Two Dimensions:
$
\begin{aligned}
& \overrightarrow{O P}=x \hat{i}+y \hat{j} \\
& ⇒|\overrightarrow{O P}|=\sqrt{x^2+y^2}
\end{aligned}
$

Three Dimension:
$\overrightarrow{OP}= x\hat{i}+y\hat{j}+z\hat{k}$
$⇒ \overrightarrow{OP}=\sqrt{x^2+y^2+z^2}$

Vectors joining two Points:
$P_1\left(x_1, y_1, z_1\right)$ and $\mathrm{P}_2\left(x_2, y_2, z_2\right)$ are any points on the axis.
Vector joining $P_1$ and $\mathrm{P}_2$ is given below:
$
\begin{aligned}
& \overrightarrow{P_1 P_2}=\left(x_2-x_1\right) \hat{i}+\left(y_2-y_1\right) \hat{j}+\left(z_2-z_1\right) \hat{k} \\
&⇒ \left|\overrightarrow{P_1 P_2}\right|=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}
\end{aligned}
$

Section Formula: $\overrightarrow{O P}$ be a position vector and line segment $A B$
- Divided internally in m:n ratio, then the formula is $\overrightarrow{O P}=\frac{m \vec{b}+n \vec{a}}{m+n}$
- Divided externally in the m:n ratio, the formula is $\overrightarrow{O P}=\frac{m \vec{b}-n \vec{a}}{m-n}$

Midpoints of vectors: $\overrightarrow{O P}=\frac{\vec{a}+\vec{b}}{2}$

Dot (scalar) Product

If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are two non-zero vectors, then their scalar product (or dot product) is denoted by $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}$ and is defined as

Observations:

1. $\quad \vec{a} \cdot \vec{b}$ is a real number.
2. $\quad \vec{a} \cdot \vec{b}$ is positive if $\theta$ is acute.
3. $\quad \vec{a} \cdot \vec{b}$ is negative if $\theta$ is obtuse.
4. $\vec{a} \cdot \vec{b}$ is zero if $\theta$ is $90^{\circ}$.
5. $\quad \vec{a} \cdot \vec{b} \leq|\vec{a}||\vec{b}|$

For any two non-zero vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, then $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=0$ if and only if $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ perpendicular to each other. i.e.
$
\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=0 \Leftrightarrow \overrightarrow{\mathbf{a}} \perp \overrightarrow{\mathbf{b}}
$

As $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are mutually perpendicular unit vectors along the coordinate axes, therefore,
$
\hat{\mathbf{i}} \cdot \hat{\mathbf{j}}=\hat{\mathbf{j}} \cdot \hat{\mathbf{i}}=0, \hat{\mathbf{j}} \cdot \hat{\mathbf{k}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{j}}=0 ; \hat{\mathbf{k}} \cdot \hat{\mathbf{i}}=\hat{\mathbf{i}} \cdot \hat{\mathbf{k}}=0
$

If $\theta=0$, then $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=|\vec{a}||\vec{b}|$
In particular, $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}}=|\overrightarrow{\mathbf{a}}|^2$
As $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are unit vectors along the coordinate axes, therefore
$
\hat{\mathbf{i}} \cdot \hat{\mathbf{i}}=|\hat{\mathbf{i}}|^2=1, \hat{\mathbf{j}} \cdot \hat{\mathbf{j}}=|\hat{\mathbf{j}}|^2=1 \text { and } \hat{\mathbf{k}} \cdot \hat{\mathbf{k}}=|\hat{\mathbf{k}}|^2=1
$

Properties of Dot (Scalar) Product

1. $\quad \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{a}} \quad$ ( commutative )
2. $\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})=\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}} \quad$ (distributive)
3. $\quad(m \overrightarrow{\mathbf{a}}) \cdot \overrightarrow{\mathbf{b}}=m(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})=\overrightarrow{\mathbf{a}} \cdot(m \overrightarrow{\mathbf{b}}) ;$ where $m$ is a scalar and $\vec{a}, \vec{b}$ are any two vectors
4. $\quad(l \overrightarrow{\mathbf{a}}) \cdot(m \overrightarrow{\mathbf{b}})=\operatorname{lm}(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})$; where $l$ and $m$ are scalars

For any two vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, we have
(i)
$
\begin{aligned}
|\vec{a} \pm \vec{b}|^2 & =|\vec{a} \pm \vec{b}| \cdot|\vec{a} \pm \vec{b}| \\
& =|\vec{a}|^2+|\vec{b}|^2 \pm 2 \vec{a} \cdot \vec{b} \\
& =|\vec{a}|^2+|\vec{b}|^2 \pm 2|\vec{a}||\vec{b}| \cos \theta
\end{aligned}
$
(ii) $|\vec{a}+\vec{b}| \cdot|\vec{a}-\vec{b}|=|\vec{a}|^2-|\vec{b}|^2$
(iii) $|\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}| \Rightarrow \vec{a}$ and $\vec{b}$ are like vectors
(iv) $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}| \Rightarrow \vec{a} \perp \vec{b}$

• 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 $\vec{a}, \vec{b}$ are non-zero, the 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 $\theta=0,$ projection of $\overrightarrow{A B}$ will be $\overrightarrow{A B}$ only.
If $\theta=\pi,$ projection of $\overrightarrow{A B}$ will be $\overrightarrow{B A}$.
If $\theta=\frac{\pi}{2}$ or $\frac{3 \pi}{2},$ projection of $\overrightarrow{A B}$ will be zero.

Cross Product:
$\theta$ is the angle between two non-parallel vectors $\vec{a}$ and $\vec{b}$, 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}=(|\vec{a}||\vec{b}| \sin \theta) \hat{n}
$
where $\hat{n}$ is the unit vector orthogonal to both $\hat{a}$ and $\hat{b}$ following the right-hand rule for direction, and $\theta$ is angle between the vectors.
$
\begin{aligned}
& \text {If } \vec{a}=\left[a_1, a_2, a_3\right] \text { and } \vec{b}=\left[b_1, b_2, b_3\right] \text { then } \\
& \vec{a} \times \vec{b}=\left[a_2 b_3-a_3 b_2, a_3 b_1-a_1 b_3, a_1 b_2-a_2 b_1\right]
\end{aligned}
$

The magnitude of the cross product vector is:
$
|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta
$
which also calculates the area of the parallelogram defined by $\vec{a}$ and $\vec{b}$
The angle $\theta$, between two vectors, $\vec{a}$ and $\vec{b}$ is given by:
$
\theta=\sin ^{-1}\left(\frac{|\vec{a}| \times|\vec{b}|}{|\vec{a}| \cdot|\vec{b}|}\right)
$

For any vectors $\vec{a}, \vec{v}$ and $\vec{w}$ and scalar $\mathrm{k} \in \mathbb{R}$
$
\begin{aligned}
& \vec{u} \times \vec{v}=-(\vec{v} \times \vec{u}) \\
& \vec{u} \times(\vec{v}+\vec{w})=\vec{u} \times \vec{v}+\vec{u} \times \vec{w} \\
& (\vec{u}+\vec{v}) \times \vec{w}=\vec{u} \times \vec{w}+\vec{v} \times \vec{w}
\end{aligned}
$

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

The link for the NCERT textbook pdf is given below:

URL: NCERT BOOK PDF

Importance of NCERT Class 12 Maths Chapter 10 Notes:

NCERT Class 12 Maths chapter 10 notes are very handy for students who aspire to achieve good marks in the board exams as well as in competitive exams. Here are some reasons why students should read these notes.

• These notes are well-structured and cover all the important concepts and formulas.
• The 2025-26 Class 12 CBSE Mathematics Syllabus has been followed to help students learn the topics that are required to get good grades on the exam.
• These notes will give conceptual clarity to students, and students can read these notes for revision.
• Completing these notes will boost the confidence of the students and create a positive vibe, which is a necessity before an exam.
• Experts who have multiple years of experience in these topics have created these notes so that students can learn these concepts easily.

Subject Wise NCERT Exemplar Solutions

After completing the NCERT textbooks, students should practice exemplar exercises for a better understanding of the chapters and clarity. The following links will help students to find exemplar exercises.

• NCERT Exemplar Class 12 Solutions

• NCERT Exemplar Class 12 Maths

• NCERT Exemplar Class 12 Physics

• NCERT Exemplar Class 12 Chemistry

• NCERT Exemplar Class 12 Biology

Subject Wise NCERT Solutions

These are links to the solutions of other subjects, which students can check to revise and strengthen those concepts.

• NCERT Solutions for Class 12 Mathematics

• NCERT Solutions for Class 12 Chemistry

• NCERT Solutions for Class 12 Physics

• NCERT Solutions for Class 12 Biology

NCERT Books and Syllabus

Students should always check the latest NCERT syllabus before planning their study routine. Also, some reference books should be read after completing the textbook exercises. The following links will be very helpful for students for these purposes.