# Problem with calculation of dot product of two vectors

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?

• Do you know how to compute dot products using the coordinates of the vectors? – Carl Christian Oct 31 '19 at 22:07
• Yes, but the problem is that the coordinates are not given. – Yan Zhuang Oct 31 '19 at 22:09
• 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? – Yan Zhuang Oct 31 '19 at 22:23
• 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. – Ned Oct 31 '19 at 22:29
• @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 – 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, $$| \vec u|^2+\vec u \cdot \vec v=0$$

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

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

• 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 – Yan Zhuang Oct 31 '19 at 22:47
• It would appear your answer key is wrong as long as you presented the question exactly right. Check. – Deepak Oct 31 '19 at 22:49
• Okay. I will write to the professor to verify! Thank you very much! – Yan Zhuang Oct 31 '19 at 22:51
• 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. – Deepak Oct 31 '19 at 22:55
• Or the answer key swapped the norms of the two vectors. – amd Oct 31 '19 at 22:56