mathematics-class-12-unit-10-chapter-06-vectors-cbcteliiz8m

### <Success style="font-size:25px"> Vectors L - 6 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- (Triangle law)
- $\overrightarrow{A A}_1=\overrightarrow{A G}+\overrightarrow{G G}_1+\vec{G}_1 \vec{A}_1 $

- $\overrightarrow{B B}_1=\overrightarrow{B G}+\overrightarrow{G G_1}+\overrightarrow{G_1 \vec{B}_1}$
- $\text { (Triangle law) } $
- $ \overrightarrow{C C}_1=\overrightarrow{C G}+\overrightarrow{G G}_1+\vec{G}_1 \vec{C}_1 $
- $ \overrightarrow{A A}_1+\overrightarrow{B B}_1+\overrightarrow{C C}_1=\overrightarrow{A G}+\overrightarrow{G G_1}+\vec{G}_1 \vec{A}_1 +\overrightarrow{B G}+\overrightarrow{G G_1}+\overrightarrow{G_1 \vec{B}_1} $
- $ +\overrightarrow{C G}+\overrightarrow{G G_1}+\overrightarrow{G_1 C_1}  =3 \overrightarrow{G G}_1+(\overrightarrow{A G}+\overrightarrow{B G}+\overrightarrow{C G}) +(\overrightarrow{G_1 A_1}+\overrightarrow{G_1 B_1}+ \overrightarrow{G_1 C_1})  =3 \overrightarrow{G G}_1+\left(\overrightarrow{A G}+\frac{2 \overrightarrow{D G}}{\overrightarrow{G A}}\right)+\left(\overrightarrow{G_1 A_1}+ 2\vec{G}_1 \vec{D}_1\right)$

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<footer style = "font-size: 18px">Vectors lecture-6 $\rightarrow$ Problem-1 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Problem-2 </footer>

### <Success style="font-size:25px"> Vectors L - 6 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $ =3 \overrightarrow{G G}_1+\left(\overrightarrow{A G}+2 \overrightarrow{D \vec{G}_1}\right)+\left(\vec{G}_1 \vec{A}_1+2 \vec{G}_1 \vec{D}_1\right) $
- $ =3 \overrightarrow{G G_1}+(\overrightarrow{A G}+\overrightarrow{G A})+\left(\overrightarrow{G_1 \vec{A}_1}+\overrightarrow{A_1 \vec{G}_1}\right) $
- $ =3 \overrightarrow{G G}_1+\overrightarrow{0}+\overrightarrow{0}=3 \overrightarrow{G G}_1=\text { RHS }$
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<footer style = "font-size: 18px">Problem-1 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-2 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Vectors L - 6 </Success>
### <primary style="font-size:24px"> Problem-2 </primary>
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- Let $P, Q, R \\& S$ be the points on the plane with position vectors $-2 i-j, 4 i, 3 i+3 j$ & $-3 i+2 j$ respectively. What kind of quadrilateral is $P Q R S$ ?

- Solution:

- $ \overrightarrow{P Q}=\overrightarrow{O Q}-\overrightarrow{O P}=(4+2) i+j=6 i+j$

- $\overrightarrow{Q R}=\overrightarrow{O R}-\overrightarrow{O Q}=(-1) i+3 j=-i+3 j $

- $ \overrightarrow{R S}=\overrightarrow{O S}-\overrightarrow{O R}=-6 i-j=-\overrightarrow{P Q} $

- $ \overrightarrow{S P}=\overrightarrow{O P}-\overrightarrow{O S}=i-3 j=-\overrightarrow{Q R} $

- $ \overrightarrow{P Q}\|\overrightarrow{R S} \quad \overrightarrow{Q R}\| \overrightarrow{S P} $
- $ \quad \text { unlike vectors }$
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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem-2 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Vectors L - 6 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- Rectangle X

- $|\overrightarrow{Q R}|=\sqrt{1+9}$
- $=\sqrt{10} \neq \sqrt{37}=|\overrightarrow{P Q}| $ Rhombus  X
- $\square$ P Q R S must be a $11^m$.
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<footer style = "font-size: 18px">Problem-2 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-3 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Vectors L - 6 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $\vec{a}+\vec{b}=5 i+2 j-3 k=\vec{c} $
- $ \vec{a} \cdot \vec{b}=6-15-10=-19 $
- $\vec{a} \cdot \vec{c}=15+10-6=19 $
- $\vec{b} \cdot \vec{c}=10-6+15=19 $
- $|\vec{a}|^2=9+25+4=38 $
- $|\vec{b}|^2=38=|\vec{c}|^2$
- $ \rightarrow \cos \theta_1=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}=\frac{-19}{38}=-\frac{1}{2}=\cos \left(120^{\circ}\right) $
- $ \left(180-\theta_1\right)=60^{\circ} $
- $ \cos \theta_2=\frac{1}{2}=\cos \theta_3 \quad \theta_2=60^{\circ}=\theta_3 $

- This must he an equilateral $\Delta$ .
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<footer style = "font-size: 18px">Solution $\rightarrow$ Problem-3 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-4 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Vectors L - 6 </Success>
### <primary style="font-size:24px"> Problem-4 </primary>
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- Show that the points $A(-1,6,6), B(-4,9,6), C(0,7,10)$ form an isosceles right angled triangle.

- Solution:
- $\overrightarrow{A B}=-3 i+3 j $

- $\overrightarrow{A C}=i+j+4 k$

- $ \overrightarrow{B C}=4 i-2 j+4 k $

- $ \overrightarrow{A B}+\overrightarrow{B C}=\overrightarrow{A C} \text { (Triangle) } $
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<footer style = "font-size: 18px">Problem-3 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem-4 </span> $\rightarrow$ Solution $\rightarrow$ Problem-5 </footer>