How does one calculate the dot product between two unit vectors
How to interpret the units of the dot or cross product of two vectors? - Physics Stack Exchange
Dot product of unit vectors
linear algebra - How does the dot product "remove" unit vectors? - Mathematics Stack Exchange
if the dot product is about how much two vector point in the same direction, why is the magnitude of the vectors a factor of the dot product? Shouldn't the angle between the two vectors be enough to decide whether they point in the similar direction or not. And when taking the dot product, how do you know whether to take the bigger angle or the smaller angle between the two vectors?
If the dot product is about how much two vector point in the same direction, why is the magnitude of the vectors a factor of the dot product? Shouldn't the angle between the two vectors be enough to decide whether they point in a similar direction or not?
Any further hint about the proof of why is the dot product the sum of the product of the components? Can't really crack this nut.
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One of two questions from my homework that Iโve been struggling with. For this one I donโt even really know where to start. Iโve never really understood unit vectors with the way my highschool teacher taught it and my uni prof hasnโt gone over it because this is more so review homework from gr12
As ACuriousMind has already noted, you can geometrically interpret the length of the cross product of two vectors as the area of the parallelogram (or as twice the area of the triangle) spanned by them, and (the absolute values of) its components as the areas of the projections of that parallelogram onto the coordinate planes.
As for the dot product of two vectors, based on the law of cosines, you can interpret it as half the difference between the sum of their squares and the square of their difference:
$$\|\vec a - \vec b\|^2 = \|\vec a\|^2 + \|\vec b\|^2 - 2(\vec a \cdot \vec b).$$
In other words, taking the vectors to be two sides of a triangle, the dot product measures (half) the amount by which Pythagoras' law fails for this triangle.
Another way to geometrically interpret (the absolute value of) the dot product is as half the area of the triangle formed by rotating one of the vectors by 90ยฐ in their common plane, and then taking the resulting vectors as two sides of a triangle:
This follows from the well known dot product formula $\vec a \cdot \vec b = \|\vec a\| \|\vec b\| \cos \gamma$, where 
is the angle between 
and 
, from the triangle area formula $T = \frac12 \|\vec a'\| \|\vec b\| \sin \gamma'$, where 
is the area of the triangle formed by the vectors 
and 
and 
is the angle between them, and the fact that the angles 
and 
are complementary, and so 
.
Note the similarity with the cross product here. In fact, we always have $\|\vec a \times \vec b\| = |\vec a' \cdot \vec b|$, where 
is 
rotated by 90ยฐ in their common plane (or in any of the planes, if there are several)!
Ps. I did notice (after posting this answer) that you asked specifically about the units of the products and "not about geometric interpretations." Even so, these examples should at least show that both the dot and the cross product of two length vectors can, in fact, be meaningfully interpreted as areas, and it should therefore not be surprising that, if the original vectors have units of, say, meters, then their product will be measured in square meters.
The length of the cross product of two vectors is the area of the parallelogram spanned by them, so the square-meters are the correct unit as well as geometrically meaningful - it's really an area. The 
-component is the area of the projection of the parallelogram onto the 
-
-plane, the 
-component the area of the projection onto the 
-
-plane and the 
-component is the area of the projection onto the 
-
-plane.
The unit of the dot product is not really meaningful. It's by definition the length of the projection of the first vector onto the second times the length of the second (or vice versa), which does not straightforwardly correspond to any area. It gets units of square-meters by definition, but there is no deeper interpretation behind it I could see.
The cross product is a rather special beast, only for $3$-dimensional space. A more general version of it is the exterior product (wedge product), which maps into a different vector space (the exterior square), which happens in the case of dimension $3$ to be identifiable with the original space. Dot products, on the other hand, can be defined in any finite number of dimensions.
Hint: Suppose dot products were vectors and not scalars. What direction should they have?