I have such a question from the exercises given by the professor

$\vec{u}$ and $\vec{v}$ are two vectors on the plane and $\vec{w}=\vec{u}+\vec{v}$. We suppose that $||\vec{u}|| = 3$ and $||\vec{v}|| = 5$ and $\vec{w} \vec{u} = 0$. Please calculate $\vec{u} \vec{v}$

According to the answer key, the procedure is as following

  • $\vec{u} \vec{v}$ = $\vec{u}(\vec{w}-\vec{u})$
  • =$\vec{u}\vec{w}-\vec{u}\vec{u}$
  • =$0-||\vec{u}||^2$
  • =-9

However, from my understand, the formula of dot product should be $\vec{u} \vec{v} = ||\vec{u}||||\vec{v}||cos\theta$

So why the $cos\theta$ is not calculated here?

  • $\begingroup$ Do you know how to compute dot products using the coordinates of the vectors? $\endgroup$ – Carl Christian Oct 31 '19 at 22:07
  • $\begingroup$ Yes, but the problem is that the coordinates are not given. $\endgroup$ – Yan Zhuang Oct 31 '19 at 22:09
  • $\begingroup$ The dot product of vectors with coordinates is easy as $u_1v_1+u_2v_2$. However, that would require me actually knowing the coordinates of each vector no? $\endgroup$ – Yan Zhuang Oct 31 '19 at 22:23
  • $\begingroup$ You don't immediately know the angle between $u$ and $v$, so you can't use that formula to find the dot product. But now that you have the dot product, you could find that angle if you wanted. $\endgroup$ – Ned Oct 31 '19 at 22:29
  • $\begingroup$ @Ned I am sorry, but I did not understand what you mean? I am not trying to find the angle. The answer provided did not calculate the angle (While the dot product of geometric vector requires the calculation of angle). So I am not sure why, in this question, they did not calculate and get the answer for the dot product $\endgroup$ – Yan Zhuang Oct 31 '19 at 22:40

You are given $\vec w \cdot \vec u =0$.

Note $\vec w \cdot \vec u = (\vec u + \vec v) \cdot \vec u$

$= \vec u \cdot \vec u + \vec v \cdot \vec u $

(distributivity of the dot product)

$=\vec u \cdot \vec u + \vec u \cdot \vec v $

(commutativity of the dot product)

$= | \vec u|^2+\vec u \cdot \vec v$

So, $| \vec u|^2+\vec u \cdot \vec v=0$

$3^2+\vec u \cdot \vec v =0$

$\therefore \vec u \cdot \vec v = -9$

  • $\begingroup$ That is to say, the answer to this question is wrong? Because there are actually three others sub-question to this in which all of them involve the result from this first question, and all of them uses -9 $\endgroup$ – Yan Zhuang Oct 31 '19 at 22:47
  • $\begingroup$ It would appear your answer key is wrong as long as you presented the question exactly right. Check. $\endgroup$ – Deepak Oct 31 '19 at 22:49
  • $\begingroup$ Okay. I will write to the professor to verify! Thank you very much! $\endgroup$ – Yan Zhuang Oct 31 '19 at 22:51
  • $\begingroup$ Yes, you should. But first ensure your question is written correctly here as I said. Because your question stipulates wv=0, but your answer key assumes uw=0. $\endgroup$ – Deepak Oct 31 '19 at 22:55
  • $\begingroup$ Or the answer key swapped the norms of the two vectors. $\endgroup$ – amd Oct 31 '19 at 22:56

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.