mathematics-class-12-unit-10-chapter-05-vectors-4rgupwzyt30

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Recall </primary>
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- i) $\vec{a}, \vec{b}$ and $\vec{a} \times \vec{b}$ from RHS of direction.

- ii) $|\vec{a} \times \vec{b}|$ is area of the parallelogram with adjecent sides as $\vec{a}$ and $\vec{b}$

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<footer style = "font-size: 18px">Vectors lecture-5 $\rightarrow$ Product of more than 2 vectors $\rightarrow$ <span style = "color:blue"> Recall </span> $\rightarrow$ Area of the parallelogram $\rightarrow$ Area of the parallelogram </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Area of the parallelogram </primary>
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- Assume $\vec{a}, \vec{b}, \vec{c}$ are 3 non-coplanar vectors

- If $\vec{a}, \vec{b}$ \& $\vec{c}$ from RHS of directions.

- Then $\vec{c}$ must be making acute angle with $\vec{a} \times \vec{b}$

- $(\vec{a} \times \vec{b}) \cdot \vec{c}>0$

- If $\vec{a}, \vec{b}, \vec{c}$ from LHS of direction then $\vec{c}$ makes obtuse angle
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<footer style = "font-size: 18px">Product of more than 2 vectors $\rightarrow$ Recall $\rightarrow$ <span style = "color:blue"> Area of the parallelogram </span> $\rightarrow$ Area of the parallelogram $\rightarrow$ Area of the parallelogram </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Area of the parallelogram </primary>
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- If $ \vec{a}, \vec{b}, \vec{c}$ from LHS of then $\vec{c}$ makes obtuse angle with $\vec{a} \times \vec{b}$. in this case,

- $ (\vec{a} \times \vec{b}) \cdot \vec{c}<0 . $

- $(\vec{a} \times \vec{b}) \cdot \vec{c}=|\vec{a} \times \vec{b}| \underbrace{|\vec{c}| \cos \theta} $

- $=$ area of the parallelogram $x$ prop. i $\vec{c}$ along $\vec{a} \times \vec{b}$ :

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<footer style = "font-size: 18px">Recall $\rightarrow$ Area of the parallelogram $\rightarrow$ <span style = "color:blue"> Area of the parallelogram </span> $\rightarrow$ Area of the parallelogram $\rightarrow$ Volume of the parallelepiped </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Volume of the parallelepiped </primary>
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- = Volume of the parallelepiped with edges $O A, O B$ and $O C$

- of $\vec{a}, \vec{b}, \vec{c}$ from LHS of direction.

- $|(\vec{a} \times \vec{b}) \cdot \vec{c}|$ is equal to the vol.

- of the parallelepiped where edges are $O A, O B, O C$

- $\therefore(\vec{a} \times \vec{b}) \cdot \vec{c}= \pm V, \quad V$ is the volume.

- Notation: $(\vec{a} \times \vec{b}) \cdot \vec{c}=[\vec{a} \vec{b} \vec{c}]$

- $ ( \vec{a} , \vec{b} , \vec{c} )$ from RHS
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<footer style = "font-size: 18px">Area of the parallelogram $\rightarrow$ Area of the parallelogram $\rightarrow$ <span style = "color:blue"> Volume of the parallelepiped </span> $\rightarrow$ Remark $\rightarrow$ $(\vec{a} \times \vec{b}) \cdot \vec{c}$ </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Remark </primary>
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- 1. If $\vec{a}, \vec{b}, \vec{c}$ form RHS, $\vec{b}, \vec{c}, \vec{a}$ also form RHS.
- $\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right]=\left[\begin{array}{lll}\vec{b} & \vec{c} & \vec{a}\end{array}\right]=\left[\begin{array}{lll}\vec{c} & \vec{a} & \vec{b}\end{array}\right]$

- 2. $[\hat{i} \hat{j} \hat{k}]=1=[j k i]=[k i j]$ and

- $ [\hat{i} \hat{j} \hat{k}] = -1 = [j i k]=[k j i]$

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<footer style = "font-size: 18px">Area of the parallelogram $\rightarrow$ Volume of the parallelepiped $\rightarrow$ <span style = "color:blue"> Remark </span> $\rightarrow$ $(\vec{a} \times \vec{b}) \cdot \vec{c}$ $\rightarrow$ $(\vec{a} \times \vec{b}) \cdot \vec{c}$ </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> $(\vec{a} \times \vec{b}) \cdot \vec{c}$ </primary>
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- How to Calculate $(\vec{a} \times \vec{b}) \cdot \vec{c}$.

- $ \vec{a}=a_1 i+a_2 j+a_3 k $

- $ \vec{b}=b_1 i+b_2 j+b_3 k $

- $\vec{c}=c_1 i+c_2 j+c_3 k . $

- $\vec{a} \times \vec{b}=\left|\begin{array}{ccc}i & j & k \\\ a_1 & a_2 & a_3 \\\ b_1 & b_2 & b_3\end{array}\right|$
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<footer style = "font-size: 18px">Volume of the parallelepiped $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> $(\vec{a} \times \vec{b}) \cdot \vec{c}$ </span> $\rightarrow$ $(\vec{a} \times \vec{b}) \cdot \vec{c}$ $\rightarrow$ Remark </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> $(\vec{a} \times \vec{b}) \cdot \vec{c}$ </primary>
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- $= \left(a_2 b_3-b_2 a_3\right) i-j\left(a_1 b_3-a_3 b_1\right)+ $
- $ \left(a_1 b_2-b_1 a_2\right) k $

- $ (\vec{a} \times \vec{b}) \cdot \vec{c}$

- $=  c_1\left(a_2 b_3-b_2 a_3\right)-c_2\left(a_1 b_3-a_3 b_1\right) +c_3\left(a_1 b_2-b_1 a_2\right) $ =

- $\left|\begin{array}{lll}a_1 & a_2 & a_3 \\\ b_1 & b_2 & b_3 \\\ c_1 & c_2 & c_3\end{array}\right|$

- $\left[\begin{array}{lll}
\vec{a} & \vec{b} & \vec{c}
\end{array}\right]= \left[\begin{array}{lll}
\vec{b} & \vec{c} & \vec{a}
\end{array}\right]=\left[\begin{array}{lll}
\vec{c} & \vec{a} & b
\end{array}\right]
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<footer style = "font-size: 18px">Remark $\rightarrow$ $(\vec{a} \times \vec{b}) \cdot \vec{c}$ $\rightarrow$ <span style = "color:blue"> $(\vec{a} \times \vec{b}) \cdot \vec{c}$ </span> $\rightarrow$ Remark $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Remark </primary>
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-  Given $\vec{a} \vec{b}, \vec{c}$
- 1. Suppose one of them is zero vector

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

- 2. Suppose two of the $\vec{a}, \vec{b}, \vec{c}$ are parallel or coincident

- (2). Suppose two of the 0,0 , parallel or coincident Then also $(\vec{a} \times \vec{b}) \cdot \vec{c}=0$

- (3) Suppose then assumption in (1) \& (2) are not true.

- Then $\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right]=0 \Rightarrow \vec{a}, \vec{b}, \vec{c}$ are coplanar conversly,

- if $\vec{a}, \vec{b}, \vec{c}$ are coplanar, (ie) $\vec{c}=\alpha \vec{a}+\beta \vec{b}$

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<footer style = "font-size: 18px">$(\vec{a} \times \vec{b}) \cdot \vec{c}$ $\rightarrow$ $(\vec{a} \times \vec{b}) \cdot \vec{c}$ $\rightarrow$ <span style = "color:blue"> Remark </span> $\rightarrow$ Examples $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- Ex:

- [$ \vec{a}  =i+2 j-k $

- $ \vec{b}  =2 i+3 j+4 k $

- $\vec{c}  =3 i+4 j+9 k $ ] are coplanar

- $(\vec{a} \times \vec{b}) \cdot \vec{c} $

- $ =\left|\begin{array}{ccc}1 & 2 & -1 \\\ 2 & 3 & 4 \\\ 3 & 4 & 9\end{array}\right|=0$

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<footer style = "font-size: 18px">Remark $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Problem-1 $\rightarrow$ Vector triple product </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Vector triple product </primary>
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- $ (\vec{a} \times \vec{b}) \times \vec{c} ?$
- $ \text { Let } \vec{d}=(\vec{a} \times \vec{b}) \times \vec{c} $
- $ \vec{d} \perp \times(\vec{a} \times \vec{b}) \ \vec{d} \perp r \vec{c} .$

- But $\vec{a} \times \vec{b}$ is perpendicular to the plane containing $\vec{a}$ and $\vec{b}$. That means, $\vec{d}$ lies on the plane containing $\vec{a}$ and $\vec{b}$.

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

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

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

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<footer style = "font-size: 18px">Examples $\rightarrow$ Problem-1 $\rightarrow$ <span style = "color:blue"> Vector triple product </span> $\rightarrow$ Vector triple product $\rightarrow$ Vector triple product </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Vector triple product </primary>
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- $ 0=\alpha \vec{a} \cdot \vec{c}+\beta \vec{b} \cdot \vec{c} $

- $\frac{\alpha}{\vec{b} \cdot \vec{c}}=-\frac{\beta}{\vec{a} \cdot \vec{c}}(=\lambda) $

- $\Rightarrow \alpha=\lambda \vec{b} \cdot \vec{c}$ and  $\beta=-\lambda(\vec{a} \cdot \vec{c}) $

- $\therefore(\vec{a} \times \vec{b}) \times \vec{c}=\lambda[(\vec{b} \cdot \vec{c}) \vec{a}-(\vec{a} \cdot \vec{c}) \vec{b}]$
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<footer style = "font-size: 18px">Problem-1 $\rightarrow$ Vector triple product $\rightarrow$ <span style = "color:blue"> Vector triple product </span> $\rightarrow$ Vector triple product $\rightarrow$ Vector triple product </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Vector triple product </primary>
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- Claim: $-\lambda=-1$.
- Given $\vec{a}, \vec{b}, \vec{c}$

- Choose the unit vectors $i, j$ \\& $k$ mutually $\perp$  \\& form RHS as follows: 
- choose $i$ along $\vec{a}$
- (ii) $\vec{a}=a_1 i$

- choose $j$ in the plane of $\vec{a}$ and  $\vec{b}$ so that it is $\perp_r$ to $i$. So that it is Ir
- $\therefore \vec{b}=b_1 i+b_2 j$

- Choose $k$ Iv to the plane of $\vec{a} \\& \vec{b}$
- so that $i, j, k$ form RHS.
- $ \therefore \vec{c}=c_1 i+c_2 j+c_3 k $
- $(\vec{a} \times \vec{b}) \times \vec{c}=a_1 b_2 c_1 j-a_1 b_2 c_2 i $
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<footer style = "font-size: 18px">Vector triple product $\rightarrow$ Vector triple product $\rightarrow$ <span style = "color:blue"> Vector triple product </span> $\rightarrow$ Vector triple product $\rightarrow$ Problem-2 </footer>

### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Vector triple product </primary>
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- Then
- $(\vec{a} \times \vec{b}) \times \vec{c}=a_1 b_2 c_1 j-a_1 b_2 c_2 i $

- $ (\vec{b} \cdot \vec{c}) \vec{a}-(\vec{a} \cdot \vec{c}) \vec{b}=a_1 b_2 c_2 i-a_1 b_2 c_1 j $
- $ \Rightarrow \lambda=-1 $

- $ (\vec{a} \times \vec{b}) \times \vec{c}=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{b} \cdot \vec{c}) \vec{a}$

- Note  :-

- $(\vec{a} \times \vec{b}) \times \vec{c} \neq \vec{a} \times(\vec{b} \times \vec{c}) \$
- $\vec{a} \times(\vec{b} \times \vec{c})=-(\vec{b} \times \vec{c}) \times \vec{a} $

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

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### <Success style="font-size:25px"> Vectors L - 5 </Success>
### <primary style="font-size:24px"> Problem-2 </primary>
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- ( * *)= $ \frac{\vec{b}|a|^2-(\vec{a} \cdot \vec{b}) \vec{a}}{|\vec{a}|^2} $

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

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

- $=  \frac{\vec{a} \times(\vec{b} \times \vec{a})}{|\vec{a}|^2}$
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<footer style = "font-size: 18px">Vector triple product $\rightarrow$ Problem-2 $\rightarrow$ <span style = "color:blue"> Problem-2 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>