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Addition of Vectors

If a→, b→, c→...

Question

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors, then vectors $\vec{a} - \vec{b}, \vec{b} - \vec{c} and \vec{c} - \vec{a}$ from a parallelopiped whose volume is ______________.

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Solution

$for vectors \vec{a}, \vec{b} and \vec{c}, [\vec{a} - \vec{b} \vec{b} - \vec{c} \vec{c} - \vec{a}] = \{(\vec{a} - \vec{b}) \times (\vec{b} - \vec{c})\} \cdot (\vec{c} - \vec{a}) [By defination] = \{(\vec{a} - \vec{b}) \times (\vec{b} - \vec{c})\} \cdot (\vec{c} - \vec{a}) = \{(\vec{a} \times \vec{b}) - (\vec{a} \times \vec{c}) - (\vec{b} \times \vec{b}) + (\vec{b} \times \vec{c})\} \cdot (\vec{c} - \vec{a}) (By distributive law) = \{(\vec{a} \times \vec{b}) - (\vec{a} \times \vec{c}) - 0 + (\vec{b} \times \vec{c})\} \cdot (\vec{c} - \vec{a}) [∵ \vec{b} \times \vec{b} = 0] = (\vec{a} \times \vec{b}) \cdot \vec{c} - (\vec{a} \times \vec{c}) \cdot \vec{c} + (\vec{b} \times \vec{c}) \cdot \vec{c} - (\vec{a} \times \vec{b}) \cdot \vec{a} + (\vec{a} \times \vec{c}) \cdot \vec{a} - (\vec{b} \times \vec{c}) \cdot \vec{a} = (\vec{a} \times \vec{b}) \cdot \vec{c} - 0 + 0 - 0 + 0 - (\vec{b} \times \vec{c}) \cdot \vec{a} = [\vec{a} \vec{b} \vec{c}] - [\vec{a} \vec{b} \vec{c}] = 0 i . e [\vec{a} - \vec{b} \vec{b} - \vec{c} \vec{c} - \vec{a}] = 0 i . e volume of parallelopiped is 0 .$

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