Question

plz answer it is very urgent from 83 to 85
donot give wrong answer paap lagega​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that the required angle be x.

Given that,

$\rm \: |\vec{a} | = 3$

$\rm \: |\vec{b} | = 5$

$\rm \: |\vec{c} | = 7$

Now, Further given that

$\rm \: \vec{a} + \vec{b} + \vec{c} = \vec{0}$

can be rewritten as

$\rm \: \vec{a} \: + \: \vec{b} \: = \: - \: \vec{c}$

On taking dot with itself on both sides, we get

$\rm \: (\vec{a} + \vec{b} ).(\vec{a} + \vec{b} ) = ( - \vec{c} ).( - \vec{c} )$

$\rm \: \vec{a} .\vec{a} + \vec{a} .\vec{b} + \vec{b} .\vec{a} + \vec{b} .\vec{b} = \vec{c} .\vec{c}$

$\rm \: { |\vec{a} | }^{2} + \vec{a} .\vec{b} + \vec{a} .\vec{b} + { |\vec{b} | }^{2} = { |\vec{c} | }^{2}$

$\rm \: {3}^{2} +2 \vec{a} .\vec{b} + {5}^{2} = {7}^{2}$

$\rm \: 9+2 |\vec{a} | |\vec{b} | \: cosx+ 25 = 49$

$\rm \: 2 (3)(5) \: cosx+ 34 = 49$

$\rm \: 30 \: cosx= 49 - 34$

$\rm \: 30 \: cosx= 15$

$\rm \: cosx = \dfrac{15}{30} = \dfrac{1}{2}$

$\rm \: cosx =cos \dfrac{\pi}{3}$

$\rm\implies \:x = \dfrac{\pi}{3}$

So, option (c) is correct.

$\large\underline{\sf{Solution-84}}$

Given that,

$\rm \: \vec{a} + \vec{b} + \vec{c} = \vec{0}$

Taking dot with itself on both sides, we get

$\rm \: (\vec{a} + \vec{b} + \vec{c} ).(\vec{a} + \vec{b} + \vec{c} ) = 0$

$\rm \: \vec{a} .\vec{a} + \vec{a} .\vec{b} + \vec{a} .\vec{c} + \vec{b} .\vec{a} + \vec{b} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{a} + \vec{c} .\vec{b} + \vec{c} .\vec{c} = 0$

$\rm \: { |\vec{a} | }^{2} + { |\vec{b} | }^{2} + { |\vec{c} | }^{2} + 2(\vec{a} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{a} ) = 0$

$\rm \: {3}^{2} + {5}^{2} + {7}^{2} + 2(\vec{a} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{a} ) = 0$

$\rm \: 9 + 25 + 49 + 2(\vec{a} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{a} ) = 0$

$\rm \: 83 + 2(\vec{a} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{a} ) = 0$

$\rm \: 2(\vec{a} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{a} ) = - 83$

$\rm\implies \:\rm \: \vec{a} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{a} = \: - \: \dfrac{83}{2}$

So, option (b) is correct.

$\large\underline{\sf{Solution-85}}$

Let required angle be x.

Given that,

$\rm \: \vec{a} + \vec{b} + \vec{c} = \vec{0}$

can be rewritten as

$\rm \: \vec{b} + \vec{c} = \: - \: \vec{a}$

On taking dot with itself on both sides, we get

$\rm \:( \vec{b} + \vec{c}).(\vec{b} + \vec{c} ) = \: ( - \: \vec{a} ).( - \vec{a} )$

$\rm \: \vec{b} .\vec{b} + \vec{b} .\vec{c} + \vec{c} .\vec{b} + \vec{c} .\vec{c} = \vec{a} .\vec{a}$

$\rm \: { |\vec{b} | }^{2} + 2(\vec{b} .\vec{c} ) + { |\vec{c} | }^{2} = { |\vec{a} | }^{2}$

$\rm \: {5}^{2} + 2 |\vec{b} | |\vec{c} | \: cosx + {7}^{2} = {3}^{2}$

$\rm \: 25+ 2 \times 5 \times 7 \times \: cosx + 49 = 9$

$\rm \: 74 + 70 \: cosx = 9$

$\rm \: 70 \: cosx = 9 - 74$

$\rm \: 70 \: cosx = - 65$

$\rm\implies \:cosx \: = \: - \: \dfrac{13}{14}$

So, option (d) is correct.

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Basic Formula Used

$\rm \: \vec{a} .\vec{a} = { |\vec{a} | }^{2}$

$\rm \: \vec{a} .\vec{b} = \vec{b} .\vec{a}$

$\rm \: \vec{a} .\vec{b} = |\vec{a} |\:|\vec{b} | \: cos \theta$