If $\vec{a},\vec{b},\vec{c},\vec{d}$are unit vectors such that $(\vec{a}\times\vec{b}).(\vec{c}\times\vec{d})=1$ and $\vec{a}.\vec{c}=\frac{1}{2}$,then
$(A)\vec{a},\vec{b},\vec{c}$ are non-coplanar

$(B)\vec{a},\vec{b},\vec{d}$ are non-coplanar

$(C)\vec{b},\vec{d}$ are non-parallel.

$(D)\vec{a},\vec{d}$ are parallel and $\vec{b}$ and $\vec{c}$ are parallel.

My Attempt:

Book's Solution which i did not understand:
$(\vec{a}\times\vec{b}).(\vec{c}\times\vec{d})=1$ is possible only when $|\vec{a}\times\vec{b}|=|\vec{c}\times\vec{d}|=0$ and $(\vec{a}\times\vec{b})$ is parallel to $(\vec{c}\times\vec{d})$ And the correct option given is $(C)$.

I could not solve this problem after some efforts.Please help me.Thanks.

  • 1
    $\begingroup$ Is there a typo? There isn't any information about $\vec{b}$ $\endgroup$ – Nicholas Nov 10 '15 at 16:11
  • $\begingroup$ Don't you mean $|\vec{a}\times\vec{b}|=|\vec{c}\times\vec{d}|=1$? $\endgroup$ – G-man Nov 13 '15 at 6:59
  • $\begingroup$ In the book,it is written as it is i have presented here.But i seriously doubt,there is some printing mistake in the book. $\endgroup$ – Vinod Kumar Punia Nov 13 '15 at 7:02

While Michael's hint was a bit vague, it did answer the question nicely. This answer is just an elaboration on it.

Since the vectors given are all unit vectors, obviously the maximum value of $|\vec{a}\times\vec{b}|$ and $|\vec{c}\times\vec{d}|$ is unity. Then it follows that the maximum value of $(\vec{a}\times\vec{b}).(\vec{c}\times\vec{d})$ is also unity. This maximum clearly occurs when $\vec{a}\perp\vec{b}$ , $\vec{c}\perp\vec{d}$ and $\vec{a}\times\vec{b}\parallel\vec{c}\times\vec{d} $

Now you just have to use your imagination. Since the two planes that are spanned by the pairs of vectors $\vec{a},\vec{b}$ and $\vec{c},\vec{d}$ are parallel, and $\vec{a}$ is not collinear with $\vec{c}$, there is no way for $\vec{b}$ to be collinear with $\vec{d}$.


There is some information to find from $$|u.v|=|u||v|\cos\theta$$ and $$|u\times v|=|u||v|\sin\theta$$

  • $\begingroup$ While these are correct formulas, it isn't clear to me how you mean this to answer to question. This might work better as a comment, if you were confining yourself to a broad hint. $\endgroup$ – hardmath Nov 10 '15 at 16:50
  • 1
    $\begingroup$ What are the angles between a and b, between c and d, and between axb and cxd? $\endgroup$ – Empy2 Nov 10 '15 at 17:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.