Calculate inner product of two vectors

Question

I know that the title is vague, at least for what I am asking. Anyway, this is an exercise that I've been struggling with for the past two hours, it seems relatively simple on a first glance, and I've tried a method which proved to be wrong. I'll go through what I've tried, but before I do I will briefly provide an overview of the exercise:

Let $ e_1, e_2, e_3$ be a basis in space $\mathbb R$

$ u = e_1 + 2e_2 +3e_3 $ and
$ v = 2e_1 - 3e_2 + 4e_3 $

$ |e_1| = 3, |e_2| = 4, |e_3| = 2$
$ e_1 \cdot e_2 = -6 $
$ e_1 \cdot e_3 = 4 $
$ e_2 \cdot e_3 = 4$

Calculate $ u \cdot v $

so this is what I've tried doing:
From $u \cdot v = |u| |v| \cos \theta$
I tried plugging the data we are given above into the forumla:

$ \frac{u \cdot v}{|u||v|} = \cos\theta $

so one example would be: $ \frac{e_1 \cdot e_2}{|e_1| |e_2|} = \cos\theta $
$ \frac{ -6}{3 \times 4} = \frac{-1}{2}$
so $\cos\theta = -0.5$
of course the other two give me
$\cos\theta = 0.5$

I then, individually, get the length of vector u and vector v by using pythagoras:
$ |u| = \sqrt{1^2 + 2^2 + 3^3 } = \sqrt{14} $
On a first glance, this does not seem right... applying the same 'logic' to
$ |v| = \ldots = \sqrt{29} $

I then plug the data into the aforementioned formula ($u \cdot v = |u| |v| \cos\theta$ )

and basically get gibberish, it does not make any sense.

I am doing something completely wrong, that I know, I am just not sure how to use the provided data to get an answer that is $ = 0 $

I'd sincerely love to get an answer as I've been on this exercise for over two hours to no avail.

EDIT:

Thank you all! It does make sense now that I look at it, and I somehow tried to over-complicate things (which is actually a real struggle of mine). However, Thanks to your help, I managed to solve it.

Answer 1

Hint: $\;e_1 \cdot v = e_1 \cdot (2e_1 - 3e_2 + 4e_3) = 2 |e_1|^2- 3 e_1 \cdot e_2 + 4 e_1 \cdot e_3 = 2 \cdot 3^2 - 3 \cdot (-6)+4 \cdot 4\,$. Now do the same for $\,u \cdot v\,$ instead of $\,e_1 \cdot v\,$.

Answer 2

Hint:

The inner product is a bilinear form, i.e. its linear in each variable and takes values in the base field $\mathbf R$.

Answer 3

HINT

The inner product in $\mathbb{R}^n$ has these three properties:

$$u \bullet v=v \bullet u$$

$$(v_1+v_2) \bullet u=u \bullet (v_1+v_2)=(u \bullet v_1)+(u \bullet v_2)$$

$$a(u \bullet v)=(au) \bullet v=u \bullet (av),\forall a \in \mathbb{R}$$

Use them and you will get the result.