If a- i-2j-k - b- i-j-k and c- - i - j-k- A vector in the plane of a- and b- whose projection on c- is 1-3- is

The correct option is C 4i j + 4k Vector lying in the plane of a and b is r=λ #160;_ 1 a+λ #160;_ 2 b and its projection on c ...

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Byju's Answer

Standard XII

Mathematics

Family of Planes Passing through the Intersection of Two Planes

If a⃗= i+2j...

Question

If $\to a = i + 2 j + k$ , $\to b = i - j + k$ and $\to c = i + j - k$ . A vector in the plane of $\to a$ and $\to b$ whose projection on $\to c$ is $1 / \sqrt{3}$ , is

4 i - j + 4 k

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3 i + j - 3 k

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2 i + j - 2 k

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4 i + j - 4 k

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Solution

The correct option is C $4 i - j + 4 k$
Vector lying in the plane of $a$ and $b$ is
$r = λ_{1} a + λ_{2} b$ and its projection on $c$ is $\frac{1}{\sqrt{3}}$
$\Rightarrow [(λ_{1} + λ_{2}) i + (2 λ_{1} - λ_{2}) j + (λ_{1} + λ_{2}) k] . \frac{i + j + k}{\sqrt{1 + 1 + 1}} = \frac{1}{\sqrt{3}}$
$\Rightarrow 2 λ_{1} - λ_{2} = 1$ and the vector is $r = (3 λ_{1} - 1) i + j + (3 λ_{1} - 1) k$
$\Rightarrow r = - 4 i + j - 4 k$ or for $λ_{1} = - 1$
$\Rightarrow - r = 4 i - j + 4 k$

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Similar questions

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Let $\to a = i + 2 j + k, \to b = i - j + k a n d \to c = i + j - k,$ a vector in the plane $\to a a n d \to b$ whose projection on $\to c i s \frac{1}{\sqrt{3}} i s . . . . . . . .$