physics-class-11-unit-04-chapter-01-introduction-to-vector-operations-lecture-1-4-cmga2z1d6ri

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Unit Vectors </primary>
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- $\hat{\imath}, \hat{\jmath}, \hat{k} \rightarrow$ symbols for unit vectors

| Unit vector | Axis |
| -------- | -------- |
| $\hat{\imath}$    | x axis     |
| $\hat{\jmath}$    | y axis     |
| $\hat{k}$    | z axis     |

- $\vec{A}=A_x \hat{\imath}+A_y \hat{\jmath}+A_3 \hat{k}$

- $ \vec{B}=B_x \hat{\imath}+B_y \hat{\jmath}+B_3 \hat{k}$

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![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-01-introduction-to-vector-operations-lecture-1_4-cmga2z1d6ri-026-0121.2.jpg)

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<footer style = "font-size: 18px"> $\rightarrow$ Introduction to Vector Operation $\rightarrow$ <span style = "color:blue"> Unit Vectors </span> $\rightarrow$ Addition of Vector $\rightarrow$ Products of Vectors </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Addition of Vector </primary>
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- $\vec{A}+\vec{B}=\left(A_x+B_x\right) \hat{\imath}+\left(A_y+B_y\right) \hat{\jmath}$

- $\text { eg } 2 \vec{A}-3 \vec{B} \quad+\left(A_3+B_3\right) \hat{k}$

- $ =2\left(A_x \hat{\imath}+A_y \hat{\jmath}+A_z \hat{k}\right)$

- $ \quad-3\left(B_x \hat{\imath}+B_y \hat{\jmath}+B_z \hat{k}\right)$

- $=\left(2 A_x-3 B_x\right) \hat{\imath}+\left(2 A_y-3 B_y\right) \hat{\jmath}$

- $+\left(2 A_z-3 B_3\right) \hat{k}$

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<footer style = "font-size: 18px">Introduction to Vector Operation $\rightarrow$ Unit Vectors $\rightarrow$ <span style = "color:blue"> Addition of Vector </span> $\rightarrow$ Products of Vectors $\rightarrow$ Distributive Law of Multiplication </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Distributive Law of Multiplication </primary>
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- Define 2 different types of products of vectors.

- Both product satisfy distributive law of multiplication.

- i.e. product of $\vec{A}$ and $(\vec{B}+\vec{C})$

- = (product of $\vec{A}$ and $\vec{B})$ + (product of $\vec{A}$ and $\vec{C}$)

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<footer style = "font-size: 18px">Addition of Vector $\rightarrow$ Products of Vectors $\rightarrow$ <span style = "color:blue"> Distributive Law of Multiplication </span> $\rightarrow$ Scalar Product or Dot Product $\rightarrow$ Properties of Scalar Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Scalar Product or Dot Product </primary>
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- Scalar product of $\vec{A}$ $\text{and}$ $\vec{B}$ given by multiplying 3 quantities

- a) $|\vec{A}|$

- b) $|\vec{B}|$

- c) cosine $\theta$

- $ \vec{A} \cdot \vec{B}=|\vec{A}||\vec{B}| \cos \theta $

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<footer style = "font-size: 18px">Products of Vectors $\rightarrow$ Distributive Law of Multiplication $\rightarrow$ <span style = "color:blue"> Scalar Product or Dot Product </span> $\rightarrow$ Properties of Scalar Product $\rightarrow$ Properties of Scalar Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Properties of Scalar Product </primary>
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- **a) scalar product is commutative**

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

- $|\vec{A}||\vec{B}| \cos \theta $

- $|\vec{B}||\vec{A}| \cos \theta$

- **b) $\vec{A} \cdot \vec{B}$ is a scalar, it can be +ve or -ve.**

- sign will defind on $\angle \theta$.

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<footer style = "font-size: 18px">Distributive Law of Multiplication $\rightarrow$ Scalar Product or Dot Product $\rightarrow$ <span style = "color:blue"> Properties of Scalar Product </span> $\rightarrow$ Properties of Scalar Product $\rightarrow$ Orthogonality of Scalar Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Properties of Scalar Product </primary>
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- if $\theta$ between 0 and $\frac{\pi}{2}$

- $\vec{A} \cdot \vec{B}$ +ve

- $\theta$ lies $\frac{\pi}{2}$ and $\pi$

- $\vec{A} \cdot \vec{B} \rightarrow $ -ve

- a) $\vec{A} \cdot \vec{A}=|\vec{A}||\vec{A}| \cos 0^{\circ}=|\vec{A}|^2$

- b) If $\vec{A} \cdot \vec{B}=0$, then

- $|\vec{A}|=0$ or $|\vec{B}|=0$ or $\cos \theta=0$

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<footer style = "font-size: 18px">Scalar Product or Dot Product $\rightarrow$ Properties of Scalar Product $\rightarrow$ <span style = "color:blue"> Properties of Scalar Product </span> $\rightarrow$ Orthogonality of Scalar Product $\rightarrow$ Dot Product of Unit Vector </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Orthogonality of Scalar Product </primary>
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- $ \vec{A} \perp \vec{B}$

- $\vec{A}$ is orthogonal to $\vec{B}$

- Scaler product of 2 unit vectors will be cosine of angle between them.

- $\vec{A} \cdot(\vec{B}+\vec{C})=(\vec{A} \cdot \vec{B})+(\vec{A} \cdot \vec{C})$

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<footer style = "font-size: 18px">Properties of Scalar Product $\rightarrow$ Properties of Scalar Product $\rightarrow$ <span style = "color:blue"> Orthogonality of Scalar Product </span> $\rightarrow$ Dot Product of Unit Vector $\rightarrow$ Dot Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Dot Product of Unit Vector </primary>
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- unit vectors $\hat{\imath} $, $\hat{\jmath}$, $\hat{k}$

- along Cartision axes

- $ i \cdot \hat{\imath}=1 $

- $\hat{\jmath} \cdot \hat{\jmath}=1, $

- $\hat{k} \cdot \hat{k}=1 $

- $ \hat{\imath} \cdot \hat{\jmath}=0 $

- $ \hat{\imath} \cdot \hat{k}=0, $

- $ \hat{\jmath} \cdot \hat{k}=0 $

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<footer style = "font-size: 18px">Properties of Scalar Product $\rightarrow$ Orthogonality of Scalar Product $\rightarrow$ <span style = "color:blue"> Dot Product of Unit Vector </span> $\rightarrow$ Dot Product $\rightarrow$ Law of Cosines </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Dot Product </primary>
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- $ \vec{A} =A_x \hat{\imath}+A_y \hat{\jmath}+A_3 \hat{k} $

- $ \vec{B} =B_x \hat{\imath}+B_y \hat{\jmath}+B_3 \hat{k} $

- $ \vec{A} \cdot \vec{B} =\left(A_x \imath+A_y \hat{\jmath}+A_3 \hat{k}\right) \cdot\left(B_x \hat{\imath}+B_y \hat{\jmath}+B_3 \hat{k}\right) $

- $ \vec{A} \cdot \vec{B}=A_x B_x+A_y B_y+A_z B_z $

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<footer style = "font-size: 18px">Orthogonality of Scalar Product $\rightarrow$ Dot Product of Unit Vector $\rightarrow$ <span style = "color:blue"> Dot Product </span> $\rightarrow$ Law of Cosines $\rightarrow$ Dot Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Law of Cosines </primary>
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- $\vec{C}=\vec{A}-\vec{B}$

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

- $\vec{C} \cdot \vec{C}  =(\vec{A}-\vec{B}) \cdot(\vec{A}-\vec{B})$

- $\vec{C} \cdot \vec{C}=\vec{A} \cdot \vec{A}+ \vec{B} \cdot \vec{B}-2 \vec{A} \cdot \vec{B}$

- $|\vec{C}|^2=|\vec{A}|^2+|\vec{B}|^2-2|\vec{A}||\vec{B}| \cos (\theta) $

- $ C^2=A^2+B^2-2 A B \cos \theta $

- $ \cos (\vec{A}, \vec{B})=\frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|}$

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<footer style = "font-size: 18px">Dot Product of Unit Vector $\rightarrow$ Dot Product $\rightarrow$ <span style = "color:blue"> Law of Cosines </span> $\rightarrow$ Dot Product $\rightarrow$ Magnitude of Vector </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Magnitude of Vector </primary>
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- $\vec{A}=A_x \hat{\imath}+A_y \hat{\jmath}+A_z \hat{k}$

- unit vector $\hat{e_A}$ along $\vec{A}$

- $\text { unit vector: } \hat{e}_A=\frac{\vec{A}}{|\vec{A}|}$

- $|\vec{A}|=\sqrt{A_x^2+A_y^2+A_z^2}$

- $\hat{e}_A $ = $A_x \hat{\imath}$ + $A_y \hat{j}$ + $A_z \hat{k}$

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<footer style = "font-size: 18px">Law of Cosines $\rightarrow$ Dot Product $\rightarrow$ <span style = "color:blue"> Magnitude of Vector </span> $\rightarrow$ Components of Vector $\rightarrow$ Components of Vector </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Components of Vector </primary>
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- Component of A along a director $\hat{e}_B \leftarrow$  unit vector

- $\vec{A} \cdot \hat{e}_B \rightarrow$  component of  $\vec{A}$ along $\vec{B}$

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<footer style = "font-size: 18px">Dot Product $\rightarrow$ Magnitude of Vector $\rightarrow$ <span style = "color:blue"> Components of Vector </span> $\rightarrow$ Components of Vector $\rightarrow$ Components of Vector </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Components of Vector </primary>
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- component of $\vec{A} \perp \hat{e}_B$

- $\vec{A} \cdot \hat{e_B} \rightarrow$ component of $\vec{A}$ along $\hat{e}_B$

- $\left(\vec{A} \cdot \hat{e_B}\right) \hat{e_B} \rightarrow$ vector component of $\vec{A}$ along $ \hat{e_B}$

- $\vec{A}-(\vec{A} \cdot \hat{e_B}) \hat{e_B} \rightarrow$   Component of  $\vec{A} \perp \hat{e_B}$

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<footer style = "font-size: 18px">Magnitude of Vector $\rightarrow$ Components of Vector $\rightarrow$ <span style = "color:blue"> Components of Vector </span> $\rightarrow$ Components of Vector $\rightarrow$ Cross Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Cross Product </primary>
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- **Vector product of 2 vectors cross product**

- Vector product of 2 vectors is a vector.

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

- cross defined as a vector normal to the plane containing $\vec{A}$ and $\vec{B}$

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<footer style = "font-size: 18px">Components of Vector $\rightarrow$ Components of Vector $\rightarrow$ <span style = "color:blue"> Cross Product </span> $\rightarrow$ Cross Product $\rightarrow$ Cross Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Cross Product </primary>
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- rotate vector $\vec{A}$ towards $\vec{B}$ by smaller angle between $\vec{A}$ and $\vec{B}$ direction in which right handed threaded screw will move, that is the direction of $\vec{C}$.

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<footer style = "font-size: 18px">Cross Product $\rightarrow$ Cross Product $\rightarrow$ <span style = "color:blue"> Cross Product </span> $\rightarrow$ Right Hand Thumb Rule $\rightarrow$ Right Hand Thumb Rule </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Right Hand Thumb Rule </primary>
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- Place 2 vectors with tails together curl fungers of right hand in direction from $\vec{A}$ to $\vec{B}$

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<footer style = "font-size: 18px">Cross Product $\rightarrow$ Cross Product $\rightarrow$ <span style = "color:blue"> Right Hand Thumb Rule </span> $\rightarrow$ Right Hand Thumb Rule $\rightarrow$ Parallel Vectors </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Parallel Vectors </primary>
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- $ \vec{A} \times(\vec{B}+\vec{C})=(\vec{A} \times \vec{B})+(\vec{A} \times \vec{C}) $

- $ \vec{A} \times \vec{A}=0$

- 2 parallel vectors, $\vec{A}$ and $\vec{B}$

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<footer style = "font-size: 18px">Right Hand Thumb Rule $\rightarrow$ Right Hand Thumb Rule $\rightarrow$ <span style = "color:blue"> Parallel Vectors </span> $\rightarrow$ Three Dimension $\rightarrow$ Cross Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Three Dimension </primary>
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- contesion axes (3D) $\rightarrow$ Right Handed

- $\hat{i}, \hat{j}, \hat{k}$

- $\hat{i} \times \hat{i}=\hat{j} \times \hat{j}=k \times \hat{k}=0$

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<footer style = "font-size: 18px">Right Hand Thumb Rule $\rightarrow$ Parallel Vectors $\rightarrow$ <span style = "color:blue"> Three Dimension </span> $\rightarrow$ Cross Product $\rightarrow$ Cross Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Cross Product </primary>
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- $ \hat{\imath} \times \hat{\jmath}=\hat{k} $

- $ \hat{j} \times \hat{k}=\hat{i} $

- $ \hat{k} \times \hat{i}=\hat{j}$

- $\hat{j} \times \hat{i} =-\hat{k} $

- $\hat{i} \times \hat{k} =-\hat{j} $

- $\hat{k} \times \hat{j} =-\hat{i}$

- cyclic order $\rightarrow$ +ve

- anticyclic order $\rightarrow$ -ve

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<footer style = "font-size: 18px">Three Dimension $\rightarrow$ Cross Product $\rightarrow$ <span style = "color:blue"> Cross Product </span> $\rightarrow$ Cross Product $\rightarrow$ Applications of Cross Product </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Cross Product </primary>
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- $\vec{A}  =A_x \hat{i}+A_y \hat{j}+A_z \hat{k} $

- $\vec{B}  =B_x \hat{i}+B_y \hat{j}+B_z \hat{k} $

- $\vec{A} \times \vec{B} =(A_x \hat{i}+A_y \hat{j}+A_z \hat{k}) \times (B_x \hat{i}+B_y \hat{j}+B_z \hat{k})$

- = $(A_y B_z-A_z B_y) \hat{i}+(A_z B_x-A_x B_z) \hat{j} + (A_x B_y-A_y B_x) \hat{k}$

- = $\left|\begin{array}{ccc} \hat{\imath} & \hat{\jmath} & \hat{k} \\\ A_x & A_y & A_z \\\ B_x & B_y & B_z \end{array}\right|$

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<footer style = "font-size: 18px">Cross Product $\rightarrow$ Cross Product $\rightarrow$ <span style = "color:blue"> Cross Product </span> $\rightarrow$ Applications of Cross Product $\rightarrow$ Parallelopiped </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Applications of Cross Product </primary>
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- $\text { a) } |\vec{A} \times \vec{B}|=|\vec{A}||\vec{B}| \sin (\angle A, B) $

- area of parallelogram with sides $\vec{A}$ and $\vec{B}$

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<footer style = "font-size: 18px">Cross Product $\rightarrow$ Cross Product $\rightarrow$ <span style = "color:blue"> Applications of Cross Product </span> $\rightarrow$ Parallelopiped $\rightarrow$ Torque </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Parallelopiped </primary>
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- **Parallelopiped , A,B,C**

- Volumne of parallelopiped  = V

- $V = (\vec{A} \times \vec{B}) \cdot \vec{C}  = (\vec{B} \times \vec{C}) \cdot \vec{A}$

- = $ (\vec{C} \times \vec{A} ) \cdot \vec{B} $
 
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<footer style = "font-size: 18px">Cross Product $\rightarrow$ Applications of Cross Product $\rightarrow$ <span style = "color:blue"> Parallelopiped </span> $\rightarrow$ Torque $\rightarrow$ Moment of Force </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- (a) $ \vec{B} \perp \vec{D} \rightarrow 5 \hat{i}+6 \hat{j} $

- $ \vec{B} \perp \vec{D} \Rightarrow \vec{B} \cdot \vec{D}=0 $

- $ \Rightarrow 5 x+18=0 \Rightarrow x=-\frac{18}{5} \$

- 10 + 6 y = 0

- $ y=\frac{-10}{6}=-\frac{5}{3} $

- $ \vec{B} = -\frac{18}{5} \hat{c}+3 \hat{j} $

- $ \vec{C}=2 \hat{c}-\frac{5}{3} \hat{j}$

- b) To show $\vec{B} \|| \vec{C}$, takes cross product.

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<footer style = "font-size: 18px">Question $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Question $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Introduction To Vector Operations L-1 </Success>
### <primary style="font-size:24px"> Question </primary>
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- $\vec{A}=5 \hat{i}+10 \hat{j}$

- $\Rightarrow$  Find component of $\vec{A}$ along $\vec{B}$ where $\vec{B}=3 \hat{i}+4 \hat{j}$

- b) Find the component of $\vec{A} \perp \vec{B}$.

- a) First find $\hat{e}_B$.

- $\vec{A} \cdot \hat{e}_B \rightarrow$ component of $\vec{A}$ along $\vec{B}$

- b) $\vec{A}-(\vec{A} \cdot \vec{B}) \vec{e}_B \leftarrow \cdot$ component of $\vec{A} \perp \vec{B}$

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<footer style = "font-size: 18px">Problem $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Question </span> $\rightarrow$ Thank You $\rightarrow$  </footer>