A vector of magnitude -2 coplanar with the vectors a -2k- and b-2-k- and perpendicular to the vector c-k- is

A vector coplanar with a and b and perpendicular to c is λ((a×b) ×c) nbsp; whose magnitude is √(2). Butλ((a×b) ×c)=λ[(a·c) b (b·c) a] =λ[4 b 4 a] (∵a·c=b·c=4) ...

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Standard XII

Mathematics

Addition of Vectors

A vector of m...

Question

A vector of magnitude $\sqrt{2}$ coplanar with the vectors $\to a =^i +^j + 2^k and \to b =^i + 2^j +^k$ and perpendicular to the vector $\to c =^i +^j +^k,$ is

- \to j + \to k

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\to i - \to k

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\to i - \to j

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\to i + \to k

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Solution

The correct option is A $- \to j + \to k$
A vector coplanar with $\to a$ and $\to b$ and perpendicular to $\to c$ is $λ ((\to a \times \to b) \times \to c)$ whose magnitude is $\sqrt{2} .$
$But λ ((\to a \times \to b) \times \to c) = λ [(\to a \cdot \to c) \to b - (\to b \cdot \to c) \to a]$
$= λ [4 \to b - 4 \to a] (∵ \to a \cdot \to c = \to b \cdot \to c = 4)$
$= 4 λ [^j -^k]$
Now, $4 | λ | \sqrt{2} = \sqrt{2}$ (Given)
$λ = \pm \frac{1}{4}$
Hence the required vector is $^j -^k$ or $-^j +^k$

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Addition of Vectors

Standard XII Mathematics

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A vector of magnitude √2 coplanar with the vectors →a=^i+^j+2^k and →b=^i+2^j+^k and perpendicular to the vector →c=^i+^j+^k, is

A vector of magnitude $\sqrt{2}$ coplanar with the vectors $\to a =^i +^j + 2^k and \to b =^i + 2^j +^k$ and perpendicular to the vector $\to c =^i +^j +^k,$ is