Dot product for Vector equation

[ricky_fusion]

Hi,

I have a question about dot product for vector.

For detail : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.4&page=theory

Is there anyone understand about it and explain to me the basic concept, why :
1. A • B = |A| |B| cosθ, not A • B =A^2 • B^2 - 2AB cosθ (Cosinus equation)
2. A|| = A cosθ u = (A • u) u (for vector component)
A|| = A • u (for direction component)

Thanks,
Ricky

 

[nasu]

Hi,

Is there anyone understand about it and explain to me the basic concept, why :
1. A • B = |A| |B| cosθ, not A • B =A^2 • B^2 - 2AB cosθ (Cosinus equation)

2. A|| = A cosθ u = (A • u) u (for vector component)
A|| = A • u (for direction component)

Thanks,
Ricky

1. For one thing, the square of a vector is a scalar. Does not make sense to have the dot product of two scalars. In the cosinus equation, the left hand side is not A*B but the third side (C). Your "alternative" does not make any sense, sorry.

The dot product can be DEFINED in several equivalent ways. One of them is the above, another way is to use components. There is not much point in asking why something is defined this way and not the other... You can ask if this is an useful operation between vectors and the answer is definitely YES.

2. A*u and A cos(theta) are the same thing if theta is the angle between A and u.
(A*u) is the projection (a scalar) of A along the unit vector u. If you multiply the projection with the vector u you get a vector, the component of A along u.