mathematics-class-12-unit-10-chapter-04-vectors-jd6qc8qgkre

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Vector product </primary>
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- <ins>	Vector Product (cross product)</ins>

- Right handed system of directions OX, OY, OZ
- $D X, O Y, O Z^{\prime}$ is left handed system.

- $O A, O B, O C$ RHS of directions.

- $O A, O B, O C^{\prime}$ LHS of directions.

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<footer style = "font-size: 18px"> $\rightarrow$ Vectors lecture - 4 $\rightarrow$ <span style = "color:blue"> Vector product </span> $\rightarrow$ Definition $\rightarrow$ Observations </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Definition </primary>
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- Let $\vec{a}$ \& $\vec{b}$ be two vectors. Let$\theta$ be the angle between them and
- $0 \leq \theta \leq \pi$.

- $\vec{a} \times \vec{b}:=|\vec{a}||\vec{b}| \hat{n} \sin \theta$

- where $\hat{n}$ is a unit vector such that at 0 ,
- $\vec{a}, \vec{b}, \hat{n}$ (in this specific order)

- $\vec{a} \cdot \vec{b}=|a||b| \cos \theta$

- form a RHS of directed

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<footer style = "font-size: 18px">Vectors lecture - 4 $\rightarrow$ Vector product $\rightarrow$ <span style = "color:blue"> Definition </span> $\rightarrow$ Observations $\rightarrow$ Geometric Integration </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Geometric Integration </primary>
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- consider $\vec{a}$ \& $\vec{b}$

- $ \text { Area of parallelogram }$       
- $ =\text { base } \times \text { height }$      
- $ =(a) \times|b| \sin \theta$       .

- $ |\vec{a} \times \vec{b}| $

- $ =\text { Area } of \text { the }$ parallelogram whose adjustment sides are $\vec{a}$ \& $\vec{b}$.

- Also,$\frac{1}{2}|\vec{a} \times \vec{b}|$ gives

- the area of $\triangle A B C$ whose adjustment sides are $\vec{a}$ \& $\vec{b}$.

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<footer style = "font-size: 18px">Definition $\rightarrow$ Observations $\rightarrow$ <span style = "color:blue"> Geometric Integration </span> $\rightarrow$ More properties $\rightarrow$ Distributive  properties with respect of addition </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> More properties </primary>
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- i) If one of $\vec{a}$ \& $\vec{b}$ is $\overrightarrow{0}, \vec{a} \times \vec{b}=\overrightarrow{0}$

- ii) If $\vec{a}$ \& $\vec{b}$ are parallel,

- $\vec{a} \times \vec{b}=\overrightarrow{0} \quad(\because \theta=0, {or}\quad  \pi)$

- iii) $\alpha \vec{a} \times \beta \vec{b}=\alpha \beta(\vec{a} \times \vec{b})=\vec{a} \times(\alpha \beta \vec{b})$

- iv) $\vec{a} \times \vec{b}=-\vec{b} \times \vec{a}$

- v) $\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$.

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<footer style = "font-size: 18px">Observations $\rightarrow$ Geometric Integration $\rightarrow$ <span style = "color:blue"> More properties </span> $\rightarrow$ Distributive  properties with respect of addition $\rightarrow$ Analytic expression of cross product </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Distributive  properties with respect of addition </primary>
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- vi)
- $|\vec{a} \times \vec{b}|^2  =|\vec{a}|^2|\vec{b}|^2 \cdot \sin ^2 \theta$

- $ =|\vec{a}|^2|\vec{b}|^2\left(1-\cos ^2 \theta\right)$

- $ =|\vec{a}|^2|\vec{b}|^2-|\vec{a}|^2|\vec{b}|^2 \cos ^2 \theta$
- $[ \vec{a} \cdot \vec{b}=|\vec{a}|| \vec{b} \mid \cdot \cos \theta ]$

- $=|\vec{a}|^2|\vec{b}|^2-(\vec{a} \cdot \vec{b})^2$

- ${\text {Vii) }} (\vec{a}+\vec{b}) \times(\vec{c}+\vec{d})$      
- $=\vec{a} \times \vec{c}+\vec{a} \times \vec{d}+\vec{b} \times \vec{c}+\vec{b} \times \vec{d}$

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<footer style = "font-size: 18px">Geometric Integration $\rightarrow$ More properties $\rightarrow$ <span style = "color:blue"> Distributive  properties with respect of addition </span> $\rightarrow$ Analytic expression of cross product $\rightarrow$ Example-1 </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Analytic expression of cross product </primary>
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- $ =  a_1 b_1 i \times i+a_1 b_2 i \times j+a_1 b_3 i \times k$
- $ +  a_2 b_1 j \times i+a_2 b_2 j \times j+a_2 b_3 j \times k$
- $ +  a_3 b_1 k \times i+a_3 b_2 k \times j+a_3 b_3 k \times k$

- $ =  a_1 b_2 k-a_1 b_3 j \ -a_2 b_1 k+a_2 b_3 c \ +a_3 b_1 j-a_3 b_2 i$

- $=  \left(a_2 b_3-a_3 b_2\right) i+\left(a_3 b_1-a_1 b_3\right) j+$

- $\left(a_1 b_2-a_2 b_1\right) k$
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<footer style = "font-size: 18px">More properties $\rightarrow$ Distributive  properties with respect of addition $\rightarrow$ <span style = "color:blue"> Analytic expression of cross product </span> $\rightarrow$ Example-1 $\rightarrow$ Example-2 </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Example-2 </primary>
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- $\vec{a} \times(\vec{b}+\vec{c})=\left|\begin{array}{ccc}i & j & k \\\ a_1 & a_2 & a_3 \\\ b_1+c_1 & b_2+c_2 & b_3+c_3\end{array}\right|$

- $\begin{aligned} & =\left|\begin{array}{ccc}i & j & k \\\ a_1 & a_2 & a_3 \\\ b_1 & b_2 & b_3\end{array}\right|+\left|\begin{array}{ccc}i & j & k \\\ a_1 & a_2 & a_3 \\\ c_1 & c_2 & c_3\end{array}\right| \\\ & =\vec{a} \times \vec{b}+\vec{a} \times \vec{c} .\end{aligned}$

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<footer style = "font-size: 18px">Example-1 $\rightarrow$ Example-2 $\rightarrow$ <span style = "color:blue"> Example-2 </span> $\rightarrow$ Remark $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- $\therefore$ The required unit vector $=\frac{8(2 i+j)}{8 \sqrt{5}}=\frac{1}{\sqrt{5}}(2 i+j)$

- Here, $ |\vec{a}|=7$ \& $|\vec{b}|=3$.

- $\sin \theta  =\frac{|\vec{a} \times \vec{b}|}{7 \times 3}=\frac{8 \sqrt{5}}{21}$

- $\theta  =\sin ^{-1}\left(\frac{8 \sqrt{5}}{21}\right)$

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<footer style = "font-size: 18px">Remark $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-4 $\rightarrow$ Example-5 </footer>

### <Success style="font-size:25px"> Vectors L - 4 </Success>
### <primary style="font-size:24px"> Example-7 </primary>
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- <ins>Ex.</ins> Given $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$

- Prove that $\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a}$.

- $ \vec{a} \times(\vec{a}+\vec{b}+\vec{c})=\overrightarrow{0}$

- $ \vec{a} \times \vec{b}+\vec{a} \times \vec{c}=0$

- $ \therefore \vec{a} \times \vec{b}=\vec{c} \times \vec{a}$

- Similarly,  $\vec{b} \times(\vec{a}+\vec{b}+\vec{c})=0$

- gives $\vec{a} \times \vec{b}=\vec{b} \times \vec{c}$

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<footer style = "font-size: 18px">Example-5 $\rightarrow$ Example-6 $\rightarrow$ <span style = "color:blue"> Example-7 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>