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Vector Component

Let a=2î+ĵ-...

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 $b$ and $c$ 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 -^j + 5^k

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both are wrong

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both are correct

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Solution

The correct option is C both are correct
Let the required vector be $a^i + b^j + c^k .$
The coplanarity condition gives $∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & - 1 1 & 1 & - 2 a & b & c \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow 1 (c + 2 b) - 2 (c + 2 a) - 1 (b - a) = 0$
$\Rightarrow - 3 a + b - c = 0$
Projection on $\to a$ gives $\frac{2 a + b - c}{\sqrt{6}} = \frac{\sqrt{2}}{\sqrt{3}}$
$\Rightarrow 2 a + b - c = 2$
$\Rightarrow 5 a = 2$ so, $a = \frac{2}{5}$
And simultaneously $b - c = \frac{6}{5}$
If the projection is taken to be negative, we get $a = - \frac{2}{5}$ and $b - c = - \frac{6}{5}$
Hence both options a and b are correct.

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