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Cross Product of Two Vectors

Let A = ai + ...

Question

Let $\to A = a^i + b^j + c^k$ be a unit vector and $\to B$ is another vector in $R^{3}$ such that $| \to A \times \to B | = 1, \to C = \frac{1}{3} (2^i + 2^j -^k)$ and $(\to A \times \to B) . \to C =$ 1, then which of the following statement(s) is/are correct?

A

If $\to A$ lies in plane x + y + z = 0, then there are exactly two choices for $\to A$

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B

If $\to A$ lies in plane x + y + z = 0, then there are exactly 4 choices for $\to A$

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C

If a, b, c $ϵ$ I, then there is no such vector $\to A$

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D

If a, b, c $ϵ$ I, then there infinitely many choices for $\to A$

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Solution

The correct options are
A
If $\to A$ lies in plane x + y + z = 0, then there are exactly two choices for $\to A$

C
If a, b, c $ϵ$ I, then there is no such vector $\to A$

$\to C = \to A \times \to B \Rightarrow \to A . \to C = 0$
$\Rightarrow 2 a + 2 b - c = 0$
For (A) and (B) $a + b + c = 0 \Rightarrow c = 0; a + b = 0$
As $a^{2} + b^{2} + c^{2} = 1 \Rightarrow ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0) (\frac{1}{\sqrt{2}}, - \frac{1}{\sqrt{2}}, 0) \end{matrix}$
For (C) and (D)
$\begin{matrix} a^{2} + b^{2} + c^{2} = 1 2 a + 2 b = c \end{matrix}}$ are not satisfied for any triplet of integers

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