The cross product won't give you the full answer. For example, it would give you the same "angle" for 10 degrees and 170 degrees, though they're clearly different angles.
Really, you can't do it without both the dot product
and the cross product (or something like it). You need to add them both together. When you do, it makes a
new kind of product called the "Clifford product" or "geometric product" -- just like when you add a real number and an imaginary number, it makes a new kind of number called a complex number.
This "Clifford product" contains the full information about how the two vectors relate, whereas its dot product and cross product "parts" each give only a piece of the puzzle. Again, this is just like with complex numbers: the real and imaginary parts give only partial information, but the real power comes when you treat the complex number as a unified entity.
Let's get specific. Write your vectors v1 and v2 in terms of an orthonormal basis, (e_x, e_y):
v_1 = x_1e_x + y_1e_y
v_2 = x_2e_x + y_2e_y
(i.e. x_1 is the x-component of vector 1, and so on.)
The angle you're looking for is just the product of the vectors v_1 and v_2, as we'll see in a moment. To multiply the vectors, we first need to learn how to multiply the basis vectors they're composed of.
The rule for multiplying two of the
same vector is simple: because they are ortho
normal, we just get 1:
e_x e_x = 1
Multiplying two
different basis vectors is where your requested orientation comes in. By "e_xe_y", we mean:
- the plane spanned by e_x and e_y, with
- the orientation turning e_x into e_y.
(Notice this last bit implies that e_xe_y = -e_ye_x.)
We're almost ready to calculate our angle. But first, remember that both the dot
and cross products are proportional to the magnitude of the vectors. If we want to get
just the angle part, we'll need to divide them both by the magnitudes. In your example, both vectors have the same magnitude: |v_1| = \sqrt{v_1 \cdot v_1} = \sqrt{2^2 + 1^2} = \sqrt{5}, and same for |v_2|.
Now, let's calculate that angle!
<br />
\begin{align}<br />
\angle(v_1,v_2) &= \frac{v_1 v_2}{|v_1||v_2|} \\<br />
&= \frac{(x_1e_x + y_1e_y) (x_2e_x + y_2e_y)}{\sqrt{5}\sqrt{5}} \\<br />
&= (x_1x_2e_x^2 + x_1y_2e_xe_y + y_1x_2e_ye_x + y_1y_2e_y^2)/5 \\<br />
&= ([x_1x_2 + y_1y_2] + [x_1y_2-x_2y_1]e_xe_y)/5<br />
\end{align}<br />
Notice the first term in square brackets looks just like the dot product, so it must be telling us \cos\theta. Similarly, the second term looks just like the cross product, so it's telling us \sin\theta. Plug in your values to get:
<br />
\begin{align}<br />
\cos\theta &= \frac{4}{5} \\<br />
\sin\theta &= - \frac{3}{5}<br />
\end{align}<br />
Solve these equations to get your oriented angle. Try it out for a few examples! (Notice that the orientation is given in terms of e_xe_y. In other words, positive angles mean you're turning in the same direction that turns e_x to e_y.)
I hope you find this helpful!
