Consider three vectors A- - -2k- -B- - -k- and C-2- -3- -4k- A vector X- of the form -A-B- - and - are numbers is perpendicular to C- The ratio of - and - is-

The correct option is A 1 : 1 The vector, X⃗ #160;= αA⃗ + βB⃗ X⃗ #160;= α (î + ĵ 2k̂ ) + #160; β (î ĵ +k̂ ) X⃗ #160;= ( α + ...

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Byju's Answer

Standard XII

Physics

Work Done as Dot Product

Consider thre...

Question

Consider three vectors $\to A =^i +^j - 2^k, \to B =^i -^j +^k$ and $\to C = 2^i - 3^j + 4^k$ . A vector $\to X$ of the form $α \to A + β \to B$ ( $α$ and $β$ are numbers) is perpendicular to $\to C$ . The ratio of $α$ and $β$ is:

1 : 1

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2 : 1

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- 1 : 1

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3 : 1

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Solution

The correct option is A $1 : 1$
The vector, $\to X = α \to A + β \to B$
$\to X = α (^i +^j - 2^k) +$ $β (^i -^j +^k)$
$\to X = (α + β)^i + (α - β)^j + (β - 2 α)^k$

As, $\to X$ $⊥$ $\to C$ $⟹$ $\to X . \to C = 0$
$∴$ $[(α + β)^i + (α - β)^j + (β - 2 α)^k] . [2^i - 3^j + 4^k] = 0$
$2 (α + β) - 3 (α - β) + 4 (- 2 α + β) = 0$
$⟹ 9 β - 9 α = 0$

Thus, $α : β = 1 : 1$

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Consider three vectors →A=^i+^j−2^k,→B=^i−^j+^k and →C=2^i−3^j+4^k. A vector →X of the form α→A+β→B (α and β are numbers) is perpendicular to →C. The ratio of α and β is:

The correct option is A 1:1The vector, →X=α→A+β→B→X=α(^i+^j−2^k)+ β(^i−^j+^k)→X=(α+β)^i+(α−β)^j+(β−2α)^kAs, →X ⊥ →C ⟹→X.→C=0∴ [(α+β)^i+(α−β)^j+(β−2α)^k].[2^i−3^j+4^k]=02(α+β)−3(α−β)+4(−2α+β)=0⟹9β−9α=0Thus, α:β=1:1

Consider three vectors $\to A =^i +^j - 2^k, \to B =^i -^j +^k$ and $\to C = 2^i - 3^j + 4^k$ . A vector $\to X$ of the form $α \to A + β \to B$ ( $α$ and $β$ are numbers) is perpendicular to $\to C$ . The ratio of $α$ and $β$ is: