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  4. If a × b = c × d and a × c = b × d, then a - d ( a ≠ d and b ≠ c ) is, parallel to

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Q. If $a \times b = c \times d$ and $a \times c = b \times d$, then $a - d$ $( a \neq d$ and $b \neq c )$ is, parallel to

Vector Algebra

Solution:

$(a-d) \times(b-c)=0$
$ \Rightarrow a \times b-a \times c-d \times b+d \times c=0$
Using $a \times b=c \times d $
and $ a \times c = b \times d $
we get, $ c \times d-b \times d-d \times b+d \times c=0 $
$ \Rightarrow d \times c+d \times b-d \times b+d \times c=0 $
(as $ a \times b=-b \times a)$
$ \Rightarrow 0=0 $
$ \therefore (a-d) \times(b-c)=0 $
which shows $(a-d)$ is parallel to $(b-c)$, when $a \neq b$ and
$b \neq c$