Can anyone tell me why?
Because they are different things with different definitions. Why would you expect them to be similar? Do you also find it strange that one is symmetric and the other anti-symmetric?
Cross product is a vector, dot product is a number.
Here is an attempt. Look up the definition of each one. Now apply each one to two 3 dimensional vectors. It will be clear. This is just how each one is defined.
I think it might be more important for you to establish the implications and relevance of the two definitions - rather than to ask "why?" (there is no answer to that question, aamof). They are both very important and useful vector operations and both produce very meaningful and useful results. Can you think , from some example that you already know about, what the vector result of a dot product tells you, and similarly, where the cross product gives a useful result? Can you think of a quantity that isn't a vector but which is the result of the interaction of two vectors? How does the 'angle between' those two vectors affect the result? (Hint: consider Work Done)
Can you think of a vector quantity that is the result of the interaction of two other vectors and how the angle is relevant to the answer? (Hint: consider an electromagnetic effect)
a scalar projection has a direction:
This should be a vector projection has a direction.
Scalar projection is a scalar just like the dot product itself.
I meant a vector projection, not a scalar projection (fixed the previous post).
$$ ((A \cdot B) \ / \ |B|) \ (B \ / \ |B|) $$
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