Let a- b- and c be non-coplanar vectors- If -a-2b 2b-c 5c-a- -a b c- then find -

[ #160;1 amp; 2 amp; 0 0 amp;2 #160; amp; 1 1 amp; 0 amp; 5 ][a,b,c]=λ[a,b,c] ⇒[1 #160; amp;2 #160; amp;0 ...

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Standard XII

Mathematics

Condition for Two Lines to be on the Same Plane

Let a, b, and...

Question

Let a, b, and c be non-coplanar vectors. If $[a + 2 b 2 b + c 5 c + a] = λ [a b c]$ , then find $λ$

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Solution

$⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 0 0 & 2 & 1 1 & 0 & 5 \end{matrix} ⎤ ⎥ ⎦ [\to a, \to b, \to c] = λ [\to a, \to b, \to c]$
$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 0 0 & 2 & 1 1 & 0 & 5 \end{matrix} ⎤ ⎥ ⎦ - λ = 0$ where $[\to a, \to b, \to c] \neq 0$
$\Rightarrow λ = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 0 0 & 2 & 1 1 & 0 & 5 \end{matrix} ∣ ∣ ∣ ∣$
$= 1 (10 - 0) - 2 (0 - 1) + 0$
$= 10 + 2 = 12$
$∴ λ = 12$

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Let a, b, and c be non-coplanar vectors. If [a+2b2b+c5c+a]=λ[abc], then find λ

⎡⎢⎣120021105⎤⎥⎦[→a,→b,→c]=λ[→a,→b,→c]⇒⎡⎢⎣120021105⎤⎥⎦−λ=0 where [→a,→b,→c]≠0⇒λ=∣∣ ∣∣120021105∣∣ ∣∣ =1(10−0)−2(0−1)+0 =10+2=12∴λ=12

Let a, b, and c be non-coplanar vectors. If $[a + 2 b 2 b + c 5 c + a] = λ [a b c]$ , then find $λ$