Let -a-b and -cbe three non-coplanar vectors- If -a-2-b 2-b-c 5-c-a-ab-c- then find -

Determine the value of λIf ⇀a,⇀b and ⇀cbe non coplanar vectors. then [⇀a bc] =|[ a_1 a_2 a_3; b_1 b_2 b_3; c_1 c_2 c_3 ]| 0.We have, ⇒[⇀a+2⇀b 2⇀b+⇀c 5⇀c+⇀a]=λ[ ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

Let ⇀a,⇀b and...

Question

Let $\overset{⇀}{a}$ , $\overset{⇀}{b}$ and $\overset{⇀}{c}$ be three non-coplanar vectors. If $[\overset{⇀}{a} + 2 \overset{⇀}{b} 2 \overset{⇀}{b} + \overset{⇀}{c} 5 \overset{⇀}{c} + \overset{⇀}{a}] = λ [\overset{⇀}{a} \vec{b} \overset{⇀}{c}]$ then find $λ$ ?

$10$

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$11$

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$12$

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$0$

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Solution

The correct option is C
$12$

Determine the value of $λ$
If $\overset{⇀}{a}$ , $\overset{⇀}{b}$ and $\overset{⇀}{c}$ be non-coplanar vectors. then $[\overset{⇀}{a} \vec{b} \vec{c}] = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}| \neq 0$ .
We have,
$\Rightarrow [\overset{⇀}{a} + 2 \overset{⇀}{b} 2 \overset{⇀}{b} + \overset{⇀}{c} 5 \overset{⇀}{c} + \overset{⇀}{a}] = λ [\overset{⇀}{a} \vec{b} \overset{⇀}{c}] \Rightarrow |\begin{matrix} 1 & 2 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 5 \end{matrix}| [\overset{⇀}{a} \vec{b} \overset{⇀}{c}] = λ [\overset{⇀}{a} \vec{b} \overset{⇀}{c}] \Rightarrow |\begin{matrix} 1 & 2 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 5 \end{matrix}| [\overset{⇀}{a} \vec{b} \overset{⇀}{c}] - λ [\overset{⇀}{a} \vec{b} \overset{⇀}{c}] = 0 \Rightarrow [|\begin{matrix} 1 & 2 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 5 \end{matrix}| - λ] [\overset{⇀}{a} \vec{b} \overset{⇀}{c}] = 0 \Rightarrow |\begin{matrix} 1 & 2 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 5 \end{matrix}| - λ = 0 [[\overset{⇀}{a} \vec{b} \overset{⇀}{c}] \neq 0, threevectorsarecoplanar] \Rightarrow λ = |\begin{matrix} 1 & 2 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 5 \end{matrix}|$
$∴ λ = 1 (10 - 0) - 2 (0 - 1) + 0 = 10 + 2 = 12$
Hence, option (C) is the correct answer

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Let a⇀,b⇀ and c⇀be three non-coplanar vectors. If a⇀+2b⇀2b⇀+c⇀5c⇀+a⇀=λa⇀b→c⇀ then find λ?

Let $\overset{⇀}{a}$ , $\overset{⇀}{b}$ and $\overset{⇀}{c}$ be three non-coplanar vectors. If $[\overset{⇀}{a} + 2 \overset{⇀}{b} 2 \overset{⇀}{b} + \overset{⇀}{c} 5 \overset{⇀}{c} + \overset{⇀}{a}] = λ [\overset{⇀}{a} \vec{b} \overset{⇀}{c}]$ then find $λ$ ?