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Addition of Vectors

Let α⃗ =î +...

Question

Let $\to α =^i +^j +^k, \to β =^i -^j -^k; \to γ = -^i +^j -^k$ be three vectors. A vector $\to δ$ , in the plane of $\to α$ and $\to β$ , whose projection on $\to γ$ is $\frac{1}{\sqrt{3}}$ , is given by

A

-^i - 3^j - 3^k

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B

^i - 3 - 3^k

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C

-^i + 3^j + 3^k

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D

^i + 3^j - 3^k

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Solution

The correct option is C $-^i + 3^j + 3^k$
$\to δ = \to α + n \to β$
$= (^i +^j +^k) (^i -^j -^k)$
$\to δ = (1 + n)^i + (1 - n)^j + (1 - n)^k$
$\frac{\to δ . \to γ}{| \to γ |} = \frac{1}{\sqrt{3}} \Rightarrow \frac{| - 1 - n + 1 - n - 1 + n |}{\sqrt{3}} = \frac{1}{\sqrt{3}}$
$| n + 1 | = 1 \Rightarrow n + 1 \Rightarrow n = 0$
or $n = - 2$
$\to δ = -^i + 3^j + 3^k$

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