I have this formula in my formula book:
$$cos(\alpha) = \frac{\lvert \overrightarrow u \cdot \overrightarrow v \rvert}{\lvert \overrightarrow u \rvert \cdot \lvert \overrightarrow v \rvert}$$
Also, this same formula book says:
$$ \overrightarrow u \cdot \overrightarrow b = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 $$
Which confuses me. The result of the $ \overrightarrow u \cdot \overrightarrow b = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 $ operation is in my understanding not a vector, and rather instead a single value, like $5$.
That is confusing, because my formula book says:
$$ \lvert \overrightarrow a \rvert = \sqrt{a_1^2 + a_2^2 + a_3^2 }$$
and therefore, $ \lvert \overrightarrow a \rvert = \sqrt{a_1^2 + a_2^2 + a_3^2 }$ requires a vector and not a single value like $5$.
My question: Where's my missunderstanding?
Note: My mathematical knowledge is limited, i'm on a high school education level
If $\vec{v} \in \mathbb{R}^n$ is a vector, then
$$\lvert\vec{v}\rvert = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2}$$
which is called the Euclidean norm of a vector, often denoted with double bars: $\lVert \vec{v} \rVert$.
If $a \in \mathbb{R}$ is a number, then $\lvert a \rvert$ is just the absolute value function. You can also think of this as a particular instance of the norm of a vector with $n = 1$:
$$\lVert\vec{v}\rVert = \sqrt{v_1^2} = \lvert v_1\rvert.$$