physics-class-11-unit-04-chapter-02-introduction-to-vectors-lecture-2-4-hiasjecqa64

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Magnitude of Vector </primary>
<hr>
<main>

- Magnitude of vector $\vec {v}$ = $|\vec{v}|$ or $ {v}$

- At time t, the particle is at P. At time t' prime, particle is at P'.

- Choose $x$ and $y$ mutually perpendicular direction.

- The origin represented by O.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Vector $\rightarrow$ Represent a Vector $\rightarrow$ <span style = "color:blue"> Magnitude of Vector </span> $\rightarrow$ Position Vector $\rightarrow$ Position Vector </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Position Vector </primary>
<hr>
<main>

- Line from $O P=\vec{r}$, position vector of $P$ at time $t$.

- $\vec{r}=\overrightarrow{O P}=$ position vector of $P$ at time $t$.

- $\vec{r}^{\prime}=\overrightarrow{O P^{\prime}}=$ position vector of $P$ at time $t^{\prime}$.

- $\overrightarrow{P P^{\prime}}=$ displacement vector of $P$.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Represent a Vector $\rightarrow$ Magnitude of Vector $\rightarrow$ <span style = "color:blue"> Position Vector </span> $\rightarrow$ Position Vector $\rightarrow$ Equality of Vectors </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Equality of Vectors </primary>
<hr>
<main>

- Shift vector parallel to itself, till the two tails touch.

- If the two heads coincide

- $\vec{A}=\vec{B}$

- Magnitude of $A$ and $B$ may be equal $(A=B)$ but $\vec{A} \neq \vec{B}$.

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-04-chapter-02-introduction-to-vectors-lecture-2_4-hiasjecqa64-225-1256.4.jpg)

</main>
<hr>
<footer style = "font-size: 18px">Position Vector $\rightarrow$ Equality of Vectors $\rightarrow$ <span style = "color:blue"> Equality of Vectors </span> $\rightarrow$ Scalar Multiplied by Vector $\rightarrow$ Scalar Multiplied by Vector </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Scalar Multiplied by Vector </primary>
<hr>
<main>

- Multiplication of a vector by a real number, (positive real number, $\lambda$ ).

- $\lambda$ is a scalar.

- Scalar multiplied by vector $=\lambda \vec{A}$

- Vector whose direction is same as $ \vec{A}$ but magnitude is $\lambda$ time magnitude of $A$.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Equality of Vectors $\rightarrow$ Equality of Vectors $\rightarrow$ <span style = "color:blue"> Scalar Multiplied by Vector </span> $\rightarrow$ Scalar Multiplied by Vector $\rightarrow$ Dimension of Vector </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Scalar Multiplied by Vector </primary>
<hr>
<main>

- $\lambda \vec{A}$ is a another vector in the same direction.

- $\lambda>1$, magnitude is bigger

- $\lambda<1$, magnitude is smaller

- $\vec{R}=\lambda \vec{A}$

- $|\vec{R}|=R=|\lambda \vec{A}|=\lambda|\vec{A}|=\lambda A$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Equality of Vectors $\rightarrow$ Scalar Multiplied by Vector $\rightarrow$ <span style = "color:blue"> Scalar Multiplied by Vector </span> $\rightarrow$ Dimension of Vector $\rightarrow$ Special case of multiplication </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Commutative law of vector addition </primary>
<hr>
<main>

- $ \vec{B}+\vec{A}: \quad \vec{R}_1=\vec{B}+\vec{A} $

- $ \overrightarrow{R_1}=\vec{R}; \vec{B}+\vec{A}=\vec{A}+\vec{B}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Addition of Two vectors $\rightarrow$ Addition of Two vectors $\rightarrow$ <span style = "color:blue"> Commutative law of vector addition </span> $\rightarrow$ Polygon Law of Addition $\rightarrow$ Subtraction of vectors </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Polygon Law of Addition </primary>
<hr>
<main>

- <span style="color:blue">Let 3 vector, $\vec{A}, \vec{B}, \vec{C}$</span>

- <span style="color:blue">$(\vec{A}+\vec{B})+\vec{C}$</span>

- <span style="color:green">$\vec{A}+(\vec{B}+\vec{C})$</span>

- <span style="color:green">$(\vec{A}+\vec{B})+\vec{C}$</span>

- <span style="color:green">$=\vec{A}+(\vec{B}+\vec{C})$</span>

- <span style="color:blue">Associative Property</span>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Addition of Two vectors $\rightarrow$ Commutative law of vector addition $\rightarrow$ <span style = "color:blue"> Polygon Law of Addition </span> $\rightarrow$ Subtraction of vectors $\rightarrow$ Parallelogram Law of Addition of Vectors </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Parallelogram Law of Addition of Vectors </primary>
<hr>
<main>

- Arrange $\vec{A}$ and $\vec{B}$ with tails coinciding at some point. These are two adjacent side of parallelogram.

- Diagonal $=\vec{R}=\vec{A}+\vec{B}$.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Polygon Law of Addition $\rightarrow$ Subtraction of vectors $\rightarrow$ <span style = "color:blue"> Parallelogram Law of Addition of Vectors </span> $\rightarrow$ Parallelogram Law of Addition of Vectors $\rightarrow$ Resolution of Vectors </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Parallelogram Law of Addition of Vectors </primary>
<hr>
<main>

- The parallelogram law give the same result as the triangle law of addition.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Subtraction of vectors $\rightarrow$ Parallelogram Law of Addition of Vectors $\rightarrow$ <span style = "color:blue"> Parallelogram Law of Addition of Vectors </span> $\rightarrow$ Resolution of Vectors $\rightarrow$ Resolution of Vectors </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Resolution of Vectors </primary>
<hr>
<main>

- <span style="color:blue">$\lambda(\vec{A}+\vec{B})=\lambda \vec{A}+\lambda \vec{B}$
</span>

- Two vectors $\vec{a}$ and $\vec{b}$ are non zero vectors in different direction in a plane.

- We have a third vector $\vec{A}$ in the same plane.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Parallelogram Law of Addition of Vectors $\rightarrow$ Parallelogram Law of Addition of Vectors $\rightarrow$ <span style = "color:blue"> Resolution of Vectors </span> $\rightarrow$ Resolution of Vectors $\rightarrow$ Resolution of Vectors </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Unit vector </primary>
<hr>
<main>

- A general point P whose coordinates are x, y, and z.

- Let us say, $\overrightarrow{A}=\overrightarrow{OP}$.

- drop a perpendicular from $P$ on $XZ$ plane

- $ \overrightarrow{O P^{\prime}}+\overrightarrow{{P}^{\prime}P}=\overrightarrow{O P} $

- x $\rightarrow$ x  component of $\overrightarrow{OP}$

- z $\rightarrow$ z  component of $\overrightarrow{OP}$

- $ \overrightarrow{{P}^{\prime}P}=y \text { component of } \overrightarrow{OP} $

</div>

</main>
<hr>
<footer style = "font-size: 18px">Resolution of Vectors $\rightarrow$ Unit vector $\rightarrow$ <span style = "color:blue"> Unit vector </span> $\rightarrow$ Unit vector $\rightarrow$ Unit vector </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Unit vector </primary>
<hr>
<main>

- $\overrightarrow{O P}$ make angle $\alpha$ with $x$ axis

- $\beta$ with $y$ axis

- $r$ with $z$ axis

- $ x=|\overrightarrow{O P}| \cos \alpha$= x component of  $\overrightarrow{O P}$

- $ y=|\overrightarrow{O P}| \cos \beta$=y component of  $\overrightarrow{O P}$

- $ z=|\overrightarrow{O P}| \cos \gamma$=z component of  $\overrightarrow{O P}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Unit vector $\rightarrow$ Unit vector $\rightarrow$ <span style = "color:blue"> Unit vector </span> $\rightarrow$ Unit vector $\rightarrow$ Resolution of Vector </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Resolution of Vector </primary>
<hr>
<main>

* $\vec{A}$, if unit vector along this is $\hat{n}$. 
* $\vec{A}=A \hat{n}$ or $|\vec{A}| \hat{n}$
* Resolution of vector $\vec{A}$ along $x$ and $y$ 
* $\vec{OP} = \vec{A}$
* $\vec{A}=\overrightarrow{O P}={\overrightarrow{O P_x}}+{\overrightarrow{O P_y}}$   
* $\vec{A} = A_x \hat{i} + A_y \hat{j}$

- **Where:**

* $\overrightarrow{O P_x} = A_x \hat{i}$
* $\overrightarrow{O P_y} = A_x \hat{j}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Unit vector $\rightarrow$ Unit vector $\rightarrow$ <span style = "color:blue"> Resolution of Vector </span> $\rightarrow$ Resolution of Vector $\rightarrow$ Resolution of Vector </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Resolution of Vector </primary>
<hr>
<main>

- $A_x$ and $A_y$$ \rightarrow $$x$ and $y$ components of $\vec{A}$
	
- $\vec{A} \rightarrow$ resolve it along $x, y$ as $A_x$ \& $A_y$.

- 2 ways to write $\vec{A}$ in a plane.

- a) magnitude of $|\vec{A}|$ and specify $\theta$ ( $\vec{A}$ make with $x$ axis)

- b) $A_x, A_y: \vec{A}=A_x \ell+A_y \hat{\jmath}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Unit vector $\rightarrow$ Resolution of Vector $\rightarrow$ <span style = "color:blue"> Resolution of Vector </span> $\rightarrow$ Resolution of Vector $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Introduction To Vectors L-2 </Success>
### <primary style="font-size:24px"> Resolution of Vector </primary>
<hr>
<main>

- $ A _x  =A \cos \theta $

- $ A x^2+A y^2  =A^2 \cos ^2 \theta+A^2 \sin ^2 \theta $

- $ =A^2 $

- $ A=\sqrt{A x^2+A y^2} $

- $ \tan \theta=\frac{A y}{A x}$

- $A_ x, A_ y $ can be positive and negative.

</div>
	
<div style='float: right; width:50%;max-width:450px;'>

</main>
<hr>
<footer style = "font-size: 18px">Resolution of Vector $\rightarrow$ Resolution of Vector $\rightarrow$ <span style = "color:blue"> Resolution of Vector </span> $\rightarrow$ Thank You $\rightarrow$  </footer>