The dot product is the norm of what vector (product)?

Question

Let $a$, $b$, $c$ be vectors in $\mathbb{R}^3$ that form a triangle.
(How this even makes sense formally, I don't know, since in a vector space all vectors are "glued" to the origin. In an affine space, or in $\mathbb{R}^3$ viewed as a manifold, it'd make more sense, perhaps.)

Let $a \bullet b$ be the dot product of vectors $a$ and $b$.

Let $a \times b$ be the cross product of vectors $a$ and $b$.

Let $|a|$ be the norm of vector $a$.

Now:

$$\cos[a,b] = {a \bullet b \over |a|\cdot|b|}$$

and

$$\sin[a,b] = {|a \times b| \over |a|\cdot|b|}$$

(Note. This last equation can't be 100% true because the number |a| is nonnegative for all vectors $a$.)

The real number $|a \times b|$ is just the norm of the cross product $a \times b$.

Which leads to the question:

The real number $a \bullet b$ is the norm of what vector product?

Also: can you replace the cross product $\times$ with the wedge product $\wedge$ to generalize the sine equation to higher dimensions?

Answer 1

I think I may have it (at least in $\mathbb{R}^2$). Praise the Lord Jesus Christ.

The dot product $a \bullet b$ of vectors $a,b \in \mathbb{R}^2$ is:

• the norm of the vector projection of vector $|b| \cdot a$ to vector $b$,
• the norm of the vector projection of vector $|a| \cdot b$ to vector $a$,

where $s \cdot a$ is the scalar product when $s$ is a scaler and $a$ is a vector.

In other words,

• extend $a$ to a circle of radius $|a| \cdot |b|$, and then project towards (the line defined by) $b$, or
• extend $b$ to a circle of radius $|b| \cdot |a|$, and then project towards (the line defined by) $a$.