mathematics-class-12-unit-10-chapter-02-vectors-m9rohy4oevs

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Recall </primary>
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- directed line segment : magnitude \& direction

- $ |\overrightarrow{A B}| =\text { length of vector }$

- $ =\text { magnitude of } \overrightarrow{AB}$

- equal, zero, Unit

- $\quad \overrightarrow{A C}=\vec{a}+\vec{b}$

- $\quad \overrightarrow{D B}=\vec{a}-\vec{b}$

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<footer style = "font-size: 18px"> $\rightarrow$ Vectors lecture - 2 $\rightarrow$ <span style = "color:blue"> Recall </span> $\rightarrow$ Position vector $\rightarrow$ Example 1 </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Example 1 </primary>
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- $P$ - mid point of $ A C$        
- $Q$ - mid point of $D B$.

- Show that $\overrightarrow{A B}+\overrightarrow{A D}+\overrightarrow{C B}+\overrightarrow{C D}$

- $=4 \overrightarrow{P Q}$

- $\text { L.H.S. }  =\overrightarrow{O B}-\overrightarrow{O A}+\overrightarrow{O D}-\overrightarrow{O A}+\overrightarrow{O B}-\overrightarrow{O C}+\overrightarrow{O D}-\overrightarrow{O C}$

- $ =2[\overrightarrow{O B}+\overrightarrow{O D}]-2[\overrightarrow{O A}+\overrightarrow{O C}]$

- $ =2[2 \overrightarrow{O Q}]-2[2 \overrightarrow{O P}], [\frac{\overrightarrow{O B}+\overrightarrow{O D}}{2}=\overrightarrow{O Q}] (?)$

- $ =4[\overrightarrow{O Q}-\overrightarrow{O P}]$
- $ =4 \overrightarrow{P Q}$

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<footer style = "font-size: 18px">Recall $\rightarrow$ Position vector $\rightarrow$ <span style = "color:blue"> Example 1 </span> $\rightarrow$ Line divided in lm $\rightarrow$ Direction cosines </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Line divided in lm </primary>
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- Let P \& Q be given two points

- Let $R$ divide $P Q$
- in the ratio $l: m$

- $R$ is given by$ \left(\frac{m x_1+l x_2}{m+l}, \frac{m y_1+l y_2}{m+l}, \frac{m z_1+l z_2}{m+l}\right)$

- $ \overrightarrow{O R}=\frac{1}{m+l}\left[m\left(x_1 i+y_1 j+z_1 k\right)+l\left(x_2 i+y_2 j+z_2 k\right)\right]$

- $=\frac{1}{m+l}\left[m \vec{r}_1+l \vec{r}_2\right]$

- In particular, take m=l:

- $R$ becomes mid point of $\overrightarrow {P Q}$ 
- Then

- $ \overrightarrow{O R}=\frac{1}{2}\left[\vec{r}_1+\vec{r}_2\right]$

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<footer style = "font-size: 18px">Position vector $\rightarrow$ Example 1 $\rightarrow$ <span style = "color:blue"> Line divided in lm </span> $\rightarrow$ Direction cosines $\rightarrow$ Example 2 </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Direction cosines </primary>
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- $ \triangle O B P$     
- $\cos \beta=\frac{b}{|\overrightarrow{O P}|}$        
- $\cos \gamma=\frac{c}{|\overrightarrow{O P}|}$        
- $\cos \alpha=\frac{a}{|\overrightarrow{O P}|}$

- $ l=\cos \alpha $      ,$ m=\cos \beta$        ,$ n=\cos \gamma$      
- $ \text { [l, m, n direction cosines of the point.] }$

- $ l^2+m^2+n^2=1$

- $ a=l \cdot\left|\overrightarrow{O P}\right|$      , $ b=m|\overrightarrow{O P}|$   ,$ C=n|\overrightarrow{O P}|$          
- $ [\text { a } b c \text { are called direction vats of the points }]$

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<footer style = "font-size: 18px">Example 1 $\rightarrow$ Line divided in lm $\rightarrow$ <span style = "color:blue"> Direction cosines </span> $\rightarrow$ Example 2 $\rightarrow$ Equation of a line </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Example 2 </primary>
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- $ \text  P(2,-1,2)$      
- $ |\overrightarrow{O P}|=\sqrt{4+1+4}=3$         
- $ \cos \alpha=2 / 3$          
- $ \cos \beta=-1 / 3$        
- $ \cos \gamma=2 / 3 .$

- $\therefore i \cos \alpha+j \cos \beta+k \cos \gamma$ is the unit vector along $\overrightarrow{O P}$ :

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<footer style = "font-size: 18px">Line divided in lm $\rightarrow$ Direction cosines $\rightarrow$ <span style = "color:blue"> Example 2 </span> $\rightarrow$ Equation of a line $\rightarrow$ Line passing through two points </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Equation of a line </primary>
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- i) To find the equation of the line passing through 0

- Let $\vec{u}$ is the unit vector along the line $L$.

- $\overrightarrow{O P}=\vec{r}$

- $\vec{r}=k \vec{u} , k$ is a scalar

- ii) To find equation n of $L$ passing though $P(a, b, c)$

- $ \vec{r}=\overrightarrow{O P}+\overrightarrow{P Q}$

- $ \vec{r}=\vec{d}+\alpha \vec{u}$

- where $\vec{u}$ is the unit vector along $L$.

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<footer style = "font-size: 18px">Direction cosines $\rightarrow$ Example 2 $\rightarrow$ <span style = "color:blue"> Equation of a line </span> $\rightarrow$ Line passing through two points $\rightarrow$ Example 3 </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Line passing through two points </primary>
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- iii) To find the equation of L passing through two point P & Q

- $ \vec{r}-\vec{d}=\alpha \vec{u}$

- $ \overrightarrow{\vec{r}}-\vec{a}=\alpha\left[\frac{\vec{b}-\vec{a}}{|\vec{b}-\vec{a}|}\right]$

- $ \vec{r}-\vec{b}=k\left[\frac{\vec{a}-\vec{b}}{\mid \vec{a}-\vec{b}|}\right]$

- $ \vec{r}=\frac{k \vec{a}}{|\vec{a}-\vec{b}|}+\left[1-\frac{k}{|\vec{a}-\vec{b}|}\right] \vec{b}$

- $ \vec{r}=t \vec{a}+(1-t) \vec{b} \leftarrow t= \frac{k}{|\vec{a}-\vec{b}|}$        
- $ \vec{r}=\alpha \vec{a}+\beta \vec{b}$

- where $\alpha+\beta=1$

- $\vec{r}=\vec{b}+t(\vec{a}-\vec{b})$

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<footer style = "font-size: 18px">Example 2 $\rightarrow$ Equation of a line $\rightarrow$ <span style = "color:blue"> Line passing through two points </span> $\rightarrow$ Example 3 $\rightarrow$ Example 4 </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Example 4 </primary>
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- $ p(9,1,2)$
- Find equation of the line through $P$ and parallel  to $(1,1,1)$.
- $\vec{r}=(9 i+j+2 k)+k(i+j+k) $

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<footer style = "font-size: 18px">Line passing through two points $\rightarrow$ Example 3 $\rightarrow$ <span style = "color:blue"> Example 4 </span> $\rightarrow$ Vector equation of a plane $\rightarrow$ Vector equation of a plane </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Vector equation of a plane </primary>
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- Vector equation of a plane

- <ins>Case (i)</ins>

- Passing the origin \& parallel to $\vec{a}$ \& $\vec{b}$

- $\vec{r}=\overrightarrow{O P}  =\overrightarrow{O A}+\overrightarrow{O B}$

- $ =\alpha \vec{a}+\beta \vec{b}$

- $ \vec{r}  =\alpha \vec{a}+\beta \overrightarrow{ b}$

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<footer style = "font-size: 18px">Example 3 $\rightarrow$ Example 4 $\rightarrow$ <span style = "color:blue"> Vector equation of a plane </span> $\rightarrow$ Vector equation of a plane $\rightarrow$ Vector equation of a plane </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
### <primary style="font-size:24px"> Vector equation of a plane </primary>
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- Case(ii).

- Passing through C
\& parallel  to vectors $\vec{a}$ \& $\vec{b}$

- $\vec{r}  =\overrightarrow{O C}+\overrightarrow{C P}$

- $\text { (io) } \vec{r}  =\vec{c}+\alpha \vec{a}+\beta \vec{b}$

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<footer style = "font-size: 18px">Example 4 $\rightarrow$ Vector equation of a plane $\rightarrow$ <span style = "color:blue"> Vector equation of a plane </span> $\rightarrow$ Vector equation of a plane $\rightarrow$ Thank you </footer>

### <Success style="font-size:25px"> Vectors L - 2 </Success>
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- <ins>Case (iii)</ins>

- Check in following
- The equation of  required plane is $\vec{r}=\alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}$ where $\gamma=1-\alpha-\beta$. $\vec{r}=\alpha \vec{a}+\beta \vec{b}+\gamma\vec{c}$

- $ \vec{r}=  \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}$

- $ \alpha+\beta+\gamma=1$

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<footer style = "font-size: 18px">Vector equation of a plane $\rightarrow$ Vector equation of a plane $\rightarrow$ <span style = "color:blue"> Vector equation of a plane </span> $\rightarrow$ Thank you $\rightarrow$  </footer>