Let a be a real number and - -2- - 2-a- - 10k- and - 12- - 20-ak- be three vectors- then - and - are

The correct option is D linearly independent for all a If #10;three vectors α #160; ,β #10; #160; ,γ #160; are linearly independent [ ...

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Standard XII

Mathematics

Linear Dependence and Independence of Vectors

Let a be a ...

Question

Let $a$ be a real number and $\to α =^i + 2^j, \to β = 2^i + a^j + 10^k$ and $\to γ = 12^i + 20^j + a^k$ be three vectors, then $\to α, \to β$ and $\to γ$ are

linearly dependent for all

a

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linearly independent for all

a

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linearly dependent for

a < 0

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linearly dependent for

a > 0

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Solution

The correct option is D linearly independent for all $a$
If three vectors $\to α, \to β, \to γ$ are linearly independent $[\to α \to β \to γ] = 0$

$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 0 2 & a & 10 12 & 20 & a \end{matrix} ∣ ∣ ∣ ∣ = 0$

$\Rightarrow a^{2} - 200 - 2 (2 a - 120) = 0$

$a^{2} - 4 a + 40 \neq 0$

$Δ = \sqrt{4 - 160} < 0$

Hence, $\to α, \to β, \to γ$ are linearly independent.

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Let a be a real number and →α=^i+2^j,→β=2^i+a^j+10^k and →γ=12^i+20^j+a^k be three vectors, then →α,→β and →γ are

The correct option is D linearly independent for all aIf three vectors →α,→β,→γ are linearly independent [→α→β→γ]=0 ⇒∣∣ ∣∣1202a101220a∣∣ ∣∣=0 ⇒a2−200−2(2a−120)=0 a2−4a+40≠0 Δ=√4−160<0 Hence, →α,→β,→γ are linearly independent.

Let $a$ be a real number and $\to α =^i + 2^j, \to β = 2^i + a^j + 10^k$ and $\to γ = 12^i + 20^j + a^k$ be three vectors, then $\to α, \to β$ and $\to γ$ are