mathematics-class-12-unit-10-chapter-01-vectors-3lzexeu35mk

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Vector  definition </primary>
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- Distance = Scalar

- Velocity = Vectors

- Mathematical Description.

- Magnitude \& direction.

- directed line segment is called vector.

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

- $ |\overrightarrow{A B}|  =\text { mag. of } \overrightarrow{A B} $

- Equal vectors: $|\overrightarrow{A B}|=|\overrightarrow{C D}|$ and $\overrightarrow{A B}$ and $\overrightarrow{C D}$ have same direction.
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<footer style = "font-size: 18px"> $\rightarrow$ Vectors lecture - 1 $\rightarrow$ <span style = "color:blue"> Vector  definition </span> $\rightarrow$ Opposite vectors $\rightarrow$ Addition of vectors </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Opposite vectors </primary>
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- Magnitude and direction NOT by its location

- If $|\vec{a}|=1$, unit vector.

- If $|\vec{a}|=0$, zero vector.
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<footer style = "font-size: 18px">Vectors lecture - 1 $\rightarrow$ Vector  definition $\rightarrow$ <span style = "color:blue"> Opposite vectors </span> $\rightarrow$ Addition of vectors $\rightarrow$ Subtraction of vectors </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Addition of vectors </primary>
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- Vectors are added in a particular way known as triangle law.

- $ \therefore$  $ \overrightarrow{A B} + \overrightarrow{B C} = \overrightarrow{A C} $

-  1. $ \vec{a} + \vec{b} = \vec{b} + \vec{a} $ ( Commutativity )
 
-  1. $ \vec{a} + \vec{b} = \vec{b} + \vec{a} $ ( Commutativity )
 
- If $ \vec{a} + ( \vec{b} + \vec{c} ) = ( \vec{a} + \vec{b} ) + \vec{c} $ ( Associative )
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<footer style = "font-size: 18px">Vector  definition $\rightarrow$ Opposite vectors $\rightarrow$ <span style = "color:blue"> Addition of vectors </span> $\rightarrow$ Subtraction of vectors $\rightarrow$ Multiplication by a Scalar </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Multiplication by a Scalar </primary>
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- Recall, If $|\vec{a}|=1, \vec{a}$ is a unit vector

- If $\overrightarrow{A B}$ is any vector. $\frac{\overrightarrow{A B}}{|\overrightarrow{A B}|}$ is a unit vector in the direction of $ \overrightarrow{A B}$
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<footer style = "font-size: 18px">Subtraction of vectors $\rightarrow$ Multiplication by a Scalar $\rightarrow$ <span style = "color:blue"> Multiplication by a Scalar </span> $\rightarrow$ Example 1 $\rightarrow$ Example 1 </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Example 1 </primary>
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- Find the other sides in terms

- $\vec{a}$ and $\vec{b}$
- $\overrightarrow{A C}  =\vec{a}+\vec{b} $
- $ A D \| B C $\&$ A D  =2 B C \quad \text { (Property of hexagon) } $
- $\overrightarrow{C D}  =\overrightarrow{C A}+\overrightarrow{A D} $
- $ =-(\vec{a}+\vec{b})+2 \vec{b} $
- $ =\vec{b}-\vec{a} $
- $\overrightarrow{D E}  =-\vec{a} $
- $\overrightarrow{E F}  =-\vec{b} $
- $ \overrightarrow{F A}  =\vec{a}-\vec{b} $

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<footer style = "font-size: 18px">Multiplication by a Scalar $\rightarrow$ Example 1 $\rightarrow$ <span style = "color:blue"> Example 1 </span> $\rightarrow$ Example 1 $\rightarrow$ Analytic description of vectors </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Analytic description of vectors </primary>
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- A unit vector along ox is denoted by $\vec{i}$ "i"

- A unit vector along oy is denoted by $\vec{j}$ "j"

- Almost, $k \lambda$ Along oy, lj Almost, $k \lambda$

- Along oy, lj

- $ \vec{r}=\overrightarrow{O P}=a i+b j $
- $ |\vec{r}|=|\overrightarrow{O P}|=\sqrt{a^2+b^2}$
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<footer style = "font-size: 18px">Example 1 $\rightarrow$ Example 1 $\rightarrow$ <span style = "color:blue"> Analytic description of vectors </span> $\rightarrow$ Analytic description of vectors $\rightarrow$ Example 2 </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Analytic description of vectors </primary>
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- $\overrightarrow{O P}$ is called the position vector of the point $P$

- P(a, b) . $\quad \overrightarrow{O P}=a i+b j$

- $ \overrightarrow{O P}  =\overrightarrow{O P^{\prime}}+\overrightarrow{P^{\prime} P} =a i+b_j+c k$
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<footer style = "font-size: 18px">Example 1 $\rightarrow$ Analytic description of vectors $\rightarrow$ <span style = "color:blue"> Analytic description of vectors </span> $\rightarrow$ Example 2 $\rightarrow$ Example 3 </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Example 2 </primary>
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- let $A(-1,1,4), B(8,0,0), C(5,-2,5)$

- $\overrightarrow{A B}  =\overrightarrow{O B}-\overrightarrow{O A} $

- $ =8 i-(-i+j+4 k) =9 i-j-4 k $

- $|\overrightarrow{A B}|  =\sqrt{81+1+16}=\sqrt{98} $

- $ \overrightarrow{B C}  =\overrightarrow{O C}-\overrightarrow{O B} $

- $ =5 i-2 j+5 k-8 i =-3 i-2 j+5 k $

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<footer style = "font-size: 18px">Analytic description of vectors $\rightarrow$ Analytic description of vectors $\rightarrow$ <span style = "color:blue"> Example 2 </span> $\rightarrow$ Example 3 $\rightarrow$ Example 3 </footer>

### <Success style="font-size:25px"> Vectors L - 1 </Success>
### <primary style="font-size:24px"> Example 3 </primary>
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- $\overrightarrow{C A}  =-i+3 j+5 k $

- $ |\overrightarrow{A B}|^2=1+4+36=41 $

- $ |\overrightarrow{B C}|^2=4+1+1=6 $

- $ |\overrightarrow{C A}|^2=1+9+25=35 $

- $ |\overrightarrow{A B}|^2=|\overrightarrow{B C}|^2+|\overrightarrow{C A}|^2 $

- $\therefore \triangle A B C$ is right angled

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<footer style = "font-size: 18px">Example 2 $\rightarrow$ Example 3 $\rightarrow$ <span style = "color:blue"> Example 3 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>