If a b = c d and a = b , show that a - d is parallel to b-c, wher… - Vidyadip

Question

$\text{If} \overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{c} \times \overrightarrow{d} and \overrightarrow{a} \times\overrightarrow{c} = \overrightarrow{b} \times\overrightarrow{d},$ show that $\overrightarrow{a} - \overrightarrow{d}$ is parallel to $\overrightarrow{b}-\overrightarrow{c},$ where $\overrightarrow{a} \neq\overrightarrow{d} and \overrightarrow{b}\neq\overrightarrow{c}.$

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Answer

$\text{For} \overrightarrow{a} - \overrightarrow{d} \text{parallel to} \overrightarrow{b} - \overrightarrow{c}, \bigg(\overrightarrow{a} - \overrightarrow{d}\bigg)\times\bigg(\overrightarrow{b} - \overrightarrow{c} \bigg) \text{should be equal to zero}$$\bigg(\overrightarrow{a} - \overrightarrow{d}\bigg) \times\bigg(\overrightarrow{b}- \overrightarrow{c}\bigg) = \overrightarrow{a}\times\overrightarrow{b} -\overrightarrow{a} \times\overrightarrow{c}-\overrightarrow{d} \times\overrightarrow{b} + \overrightarrow{d} \times\overrightarrow{c}$
$= \overrightarrow{a}\times\overrightarrow{b}- \overrightarrow{a} \times\overrightarrow{c} +\overrightarrow{b} \times\overrightarrow{d} - \overrightarrow{c} \times\overrightarrow{d}$
$= \overrightarrow{o} \bigg[ \therefore \overrightarrow{a}\times\overrightarrow{b} = \overrightarrow{c} \times\overrightarrow{d} and \overrightarrow{a}\times\overrightarrow{c} = \overrightarrow{b}\times\overrightarrow{d}\bigg]$
$\text{Thus }\bigg(\overrightarrow{a}- \overrightarrow{d}\bigg) \text{is parallel to }\bigg(\overrightarrow{b}- \overrightarrow{c}\bigg)$

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