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.

 

[Architeuthis Dux]

Hello Ricky,

Thank you for your question about the dot product for vector equations. The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and produces a scalar quantity. It is often used in physics and engineering to calculate work, energy, and other important quantities.

To understand why A • B = |A| |B| cosθ, let's first define what each term means. A and B are two vectors, |A| and |B| represent the magnitudes (or lengths) of those vectors, and θ is the angle between them. So, when we take the dot product of A and B, we are essentially multiplying their magnitudes and the cosine of the angle between them.

Now, why is it not A • B =A^2 • B^2 - 2AB cosθ? This is because the dot product is not a multiplication of individual components, but rather a combination of the two vectors as a whole. The formula A • B = |A| |B| cosθ takes into account the direction of the vectors, while the equation A^2 • B^2 - 2AB cosθ does not. This makes the first formula more accurate and useful in calculations involving vectors.

For your second question, A|| represents the component of vector A that is parallel to the unit vector u. The unit vector u is a vector with a magnitude of 1 and points in the same direction as the original vector A. The formula A|| = A cosθ u = (A • u) u calculates the parallel component of A by first finding the dot product of A and u, and then multiplying it by u. This formula takes into account the direction of the vectors and gives a more accurate result than the formula A|| = A • u, which only considers the magnitude of the vectors.

I hope this helps clarify the basic concepts of the dot product for vector equations. If you have any further questions, please don't hesitate to ask. Happy learning!