Let \overset{\rightarrow}{x} , \textrm{ } \overset{\rightarrow}{y} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{z} be three vectors each of magnitude \sqrt{2} and the angle between each pair of them is \frac{\pi}{3}. If \overset{\rightarrow}{a} is a nonzero vector perpendicular to \overset{\rightarrow}{x} \textrm{ }\textrm{ } and \textrm{ }\textrm{ }\textrm{ } \overset{\rightarrow}{y} \times \overset{\rightarrow}{z} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{b} is a nonzero vector perpendicular to \overset{\rightarrow}{y} and \overset{\rightarrow}{z} \times \overset{\rightarrow}{x}, then

Engineering

Mathematics

Vector Triple Product

Question

Let $\vec{x}, \vec{y} and \vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{π}{3}$ . If $\vec{a}$ is a nonzero vector perpendicular to $\vec{x} and \vec{y} \times \vec{z} and \vec{b}$ is a nonzero vector perpendicular to $\vec{y}$ and $\vec{z} \times \vec{x}$ , then

$\vec{a} \cdot \vec{b} = - (\vec{a} \cdot \vec{y}) (\vec{b} \cdot \vec{z})$

$\vec{a} = (\vec{a} \cdot \vec{y}) (\vec{y} - \vec{z})$

$\vec{a} = (\vec{a} \cdot \vec{y}) (\vec{z} - \vec{y})$

$\vec{b} = (\vec{b} \cdot \vec{z}) (\vec{z} - \vec{x})$

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$\vec{a} = λ [\vec{x} \times (\vec{y} \times \vec{z})] = λ [(\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z}]$

$\vec{a} = λ [\vec{y} - \vec{z}]$

take dot product with $\vec{y}$

$\vec{a} \cdot \vec{y} = λ [\vec{y} \cdot \vec{z} - \vec{y} \cdot \vec{z}] \Rightarrow λ = \vec{a} \cdot \vec{y}$

$∴ \vec{a} = (\vec{a} \cdot \vec{y}) (\vec{y} - \vec{z})$

${III}^{ly} \vec{b} = μ [\vec{y} \times (\vec{z} \times \vec{x})] = μ [(\vec{y} \cdot \vec{x}) \vec{z} - (\vec{y} \cdot \vec{z}) \vec{x}]$

$\vec{b} = μ (\vec{z} - \vec{x})$

take dot product with $\vec{z}, we get µ = \vec{b} \cdot \vec{z}$

$∴ \vec{b} = (\vec{b} \cdot \vec{z}) (\vec{z} - \vec{x})$

$and \vec{a} \cdot \vec{b} = - (\vec{a} \cdot \vec{y}) (\vec{b} \cdot \vec{z})$