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[Community Question] Linear-algebra: Neep help justifying a vector relation given in a question

One of our user asked:

i'm trying to do a question for which I was given the following line equations:

$\underline r = \underline a + \lambda \underline u$

$\underline r' = \underline a' + \lambda' \underline u'$

They then gave me this relationship without any justification, i've been trying to get my head around it but have not had much luck.

$\lvert \underline r-\underline r'\rvert^2\lvert \underline u \times \underline u'\rvert^2=\lvert (\underline a - \underline a') \cdot (\underline u \times \underline u')\lvert^2+\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert^2$

I know that

$\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert = \lvert \underline r-\underline r'\rvert\lvert \underline u \times \underline u'\rvert\sin \theta $

but cant get any further.


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