If ` vec axx vec b= vec cxx vec d\ a n d\ vec axx vec c= vec bxx vec d ,\ ` show that ` vec a- vec d` is parallel to ` vec b- vec c , w h e r e\ vec a!= vec d\ a n d\ vec b!= vec c`

Explanation

`vec{a}-vec{d}` will be parallel to `vec{b}-vec{c}`, if `(vec{a}-vec{d}) times(vec{b}-vec{c})=vec{0}`

Now

`(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} `

`=vec{a} times vec{b}-vec{a} times vec{c}+vec{b} times vec{d}-vec{c} times vec{d}`

`=0[`because given `vec{a} times vec{b}=vec{c} times vec{d} and vec{a} times vec{c}=vec{b} times vec{d}]`

Hence, `vec{a}-vec{d}| vec{b}-vec{c}`