Let a-2i-j-k-b-2-k- and c-2k- be three vectors- A vector in the plane of b- and c- whose projection on a- is of magnitude -23 is

The correct option is B 2î ĵ+5k̂ A vector in the plane of b⃗ and c⃗ #160;is equal to = b⃗ + λ #160;c⃗ = (1+λ)î +(2+λ)ĵ (1+2λ)k̂ ...

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Byju's Answer

Standard XII

Mathematics

Family of Planes Passing through the Intersection of Two Planes

Let a⃗=2i-j...

Question

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

2^i - 3^j - 3^k

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2^i + 3^j + 3^k

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- 2^i -^j + 5^k

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2^i +^j + 5^k

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Solution

The correct option is B $- 2^i -^j + 5^k$
A vector in the plane of $\to b$ and $\to c$ is equal to

$= \to b + λ \to c$

$= (1 + λ)^i + (2 + λ)^j - (1 + 2 λ)^k$

and projection of this vector on $\to a$ is equal to $λ \sqrt{\frac{2}{3}}$

$∴ \frac{(1 + λ) 2 - (2 + λ) - (1 + 2 λ)}{\sqrt{6}} = \sqrt{\frac{2}{3}}$

$∴ λ + 1 = 2$ and $- 2$

$∴ λ = 1, - 3$

Using $λ = - 3$

We get the vector as

$- 2^i -^j + 5^k$

Hence the answer is option $C .$

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Let →a=2i−j+k,→b=^i+2^j−^k and →c=^i+^j−2^k be three vectors. A vector in the plane of →b and →c whose projection on →a is of magnitude √23 is

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