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Question:

Let \overrightarrow{x}, \overrightarrow{y} and \overrightarrow{z} be three vectors each of magnitude √2 and the angle between each pair of them is π/3. If \overrightarrow{a} is a non-zero vector perpendicular to \overrightarrow{x} and \overrightarrow{y} × \overrightarrow{z} and \overrightarrow{b} is a non-zero vector perpendicular to \overrightarrow{y} and \overrightarrow{z} × \overrightarrow{x}, then which of the following is true?

\overrightarrow{a}·\overrightarrow{b} = - (\overrightarrow{a}·\overrightarrow{y})(\overrightarrow{b}·\overrightarrow{z})

\overrightarrow{a} = -(\overrightarrow{a}·\overrightarrow{y})(\overrightarrow{z} - \overrightarrow{y})

\overrightarrow{b} = (\overrightarrow{b}·\overrightarrow{z})(\overrightarrow{z} - \overrightarrow{x})

\overrightarrow{a} = (\overrightarrow{a}·\overrightarrow{y})(\overrightarrow{y} - \overrightarrow{z})

Solution:

\overrightarrow{a} is in direction of \overrightarrow{x} × (\overrightarrow{y} × \overrightarrow{z}) i.e., \overrightarrow{a} is in the direction of (\overrightarrow{x}·\overrightarrow{z})\overrightarrow{y} - (\overrightarrow{x}·\overrightarrow{y})\overrightarrow{z}. Since |\overrightarrow{x}| = |\overrightarrow{y}| = |\overrightarrow{z}| = √2 and the angle between each pair is π/3, \overrightarrow{x}·\overrightarrow{y} = \overrightarrow{y}·\overrightarrow{z} = \overrightarrow{z}·\overrightarrow{x} = 2cos(π/3) = 1. Therefore, \overrightarrow{a} is in the direction of \overrightarrow{y} - \overrightarrow{z}. Similarly, \overrightarrow{b} is in the direction of \overrightarrow{y} × (\overrightarrow{z} × \overrightarrow{x}) i.e., \overrightarrow{b} is in the direction of (\overrightarrow{y}·\overrightarrow{x})\overrightarrow{z} - (\overrightarrow{y}·\overrightarrow{z})\overrightarrow{x} = \overrightarrow{z} - \overrightarrow{x}. Now, \overrightarrow{a} = λ(\overrightarrow{y} - \overrightarrow{z}) and \overrightarrow{b} = μ(\overrightarrow{z} - \overrightarrow{x}) for some scalars λ and μ. Then \overrightarrow{a}·\overrightarrow{y} = λ(\overrightarrow{y} - \overrightarrow{z})·\overrightarrow{y} = λ(2 - 1) = λ and \overrightarrow{b}·\overrightarrow{z} = μ(\overrightarrow{z} - \overrightarrow{x})·\overrightarrow{z} = μ(2 - 1) = μ. Therefore, \overrightarrow{a}·\overrightarrow{b} = λμ(\overrightarrow{y} - \overrightarrow{z})·(\overrightarrow{z} - \overrightarrow{x}) = λμ(-1) = -(\overrightarrow{a}·\overrightarrow{y})(\overrightarrow{b}·\overrightarrow{z}). Hence option A is correct.