Let a-2-k-b-k- and c-k- - A vector in the plane of a and b- where projection on c is 1-3- is

The correct option is A 4î ĵ+4k̂ A vector in the plane of a and b is nbsp; a+λb =(î+2ĵ+k̂)+λ(î ĵ+k̂) nbsp; =(1+λ)î+(2 λ)ĵ+(1+λ)k̂ If ...

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Byju's Answer

Standard XII

Physics

Vectors

Let a=î+2ĵ+...

Question

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

4^i -^j + 4^k

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3^i +^j - 3^k

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2^i +^j

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4^i +^j - 4^k

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Solution

The correct option is A $4^i -^j + 4^k$
A vector in the plane of $\to a$ and $\to b$ is
$\to a + λ \to b = (^i + 2^j +^k) + λ (^i -^j +^k)$
$= (1 + λ)^i + (2 - λ)^j + (1 + λ)^k$
If projection on $\to c$ is $\frac{1}{\sqrt{3}}$
$\Rightarrow (\to a + λ \to b) . \frac{\to c}{∣ ∣ \to c ∣ ∣} = \pm \frac{1}{\sqrt{3}}$
$\Rightarrow ((1 + λ)^i + (2 - λ)^j + (1 + λ)^k) \frac{(^i +^j -^k)}{\sqrt{1 + 1 + 1}} = \pm \frac{1}{\sqrt{3}}$
$\Rightarrow \frac{1 + λ + 2 - λ - 1 - λ}{\sqrt{3}} = \pm \frac{1}{\sqrt{3}}$
$\Rightarrow 2 - λ = \pm 1$
$\Rightarrow λ = 2 \pm 1$ $∴ λ = 1$ or $3$
If $λ = 1, \to a + λ \to b = (1 + 1)^i + (2 - 1)^j + (1 + 1)^k$
$= 2^i +^j + 2^k$
If $λ = 3, \to a + λ \to b = (1 + 3)^i + (2 - 3)^j + (1 + 3)^k$
$= 4^i -^j + 4^k$

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Let →a=^i+2^j+^k,→b=^i−^j+^k and →c=^i+^j−^k . A vector in the plane of →a and →b, where projection on →c is 1√3, is

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