Let three vectors $\vec{a}, \vec{b} a n d \vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a} a n d \vec{b}, \vec{a} . \vec{c} = 7 a n d \vec{b}$ is perpendicular to $\vec{c}, w h e r e \vec{a} = - \hat{i} + \hat{j} + \hat{k} a n d \hat{b} = 2 \hat{i} + \hat{k},$ then the value of 2 ${| \vec{a} + \vec{b} + \vec{c} |}^{2}$ is____________.

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alok kumar singh | Contributor-Level 10

a month ago

$\vec{c} = α \vec{a} + β \vec{b} . . . . . (i)$

$\vec{a} . \vec{c} = 7 \vec{b} . \vec{c} = 0$

$\vec{a} = - \hat{i} + \hat{j} + \hat{k} \Rightarrow | \vec{a} | = \sqrt[]{3}$

$\vec{b} = 2 \hat{i} + \hat{k} \Rightarrow | \vec{b} | = \sqrt[]{5} \vec{a} . \vec{b} = - 2 + 1 = - 1$

From $(i) \vec{a} . \vec{c} = α {| \vec{a} |}^{2} - β$

$3 α - β = 7 . . . . . . . . . . (i i)$

$\bar{b} . \bar{c} = α \bar{b} . \bar{a} + β {| \vec{b} |}^{2} \Rightarrow - α + 5 β = 0 . . . . . . . (i i i)$

Solving $α = \frac{5}{2} a n d β = \frac{1}{2}$

$\Rightarrow 2 {| \vec{a} + \vec{b} + \vec{c} |}^{2} = 75$

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Let three vectors a → , b → a n d c → be such that c → is coplanar with a → a n d b → , a → . c → = 7 a n d b → is perpendicular to c → , w h e r e a → = − i ^ + j ^ + k ^ a n d b ^ = 2 i ^ + k ^ , then the value of 2 | a → + b → + c → | 2 is____________.

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