bold letters=vector

if a x b=c x d and a x c= b x d show that (a-d) is parallel to (b-c) wher a isnot =d and b isnot=c

Question

bold letters=vector
if a x b=c x d and a x c= b x
d show that (a-d) is parallel to
(b-c) wher a isnot
=d and b
isnot=c

Shridhar Kaushik · Accepted Answer

We know that two non zero vectors are parallel if and only if their cross product is a zero vector.

So, we will prove that the cross product of $(\vec{a} - \vec{d}) and (\vec{b} - \vec{c})$ is a zero vector

We have,

$(\vec{a} - \vec{d}) \times (\vec{b} - \vec{c}) = \vec{a} \times \vec{b} - \vec{a} \times \vec{c} - \vec{d} \times \vec{b} + \vec{d} \times \vec{c} \Rightarrow (\vec{a} - \vec{d}) \times (\vec{b} - \vec{c}) = \vec{c} \times \vec{d} - \vec{b} \times \vec{d} + \vec{b} \times \vec{d} - \vec{c} \times \vec{d} [∵ \vec{a} \times \vec{b} = \vec{c} \times \vec{d} and \vec{a} \times \vec{c} = \vec{b} \times \vec{d} and - \vec{d} \times \vec{b} = \vec{b} \times \vec{d}, \vec{d} \times \vec{c} = - \vec{c} \times \vec{d}] \Rightarrow (\vec{a} - \vec{d}) \times (\vec{b} - \vec{c}) = \vec{0} Hence, (\vec{a} - \vec{d}) is parallel to (\vec{b} - \vec{c})$

bold letters=vector if a x b=c x d and a x c= b x d show that (a-d) is parallel to (b-c) wher a isnot =d and b isnot=c

bold letters=vector

if a x b=c x d and a x c= b x d show that (a-d) is parallel to (b-c) wher a isnot =d and b isnot=c