vectors

[UrbanXrisis]

"...defines the dot product of two vecotors A and B as a scalar C given by the equation:

http://home.earthlink.net/~urban-xrisis/clip_image002.jpg

It wants me to show in vector notation a force of 134N towards the upper right direction 24.5 degrees from the vertical given by F={55.6x+122y}[N]

I dont understand the equation. Why does the equation have C=A*B? Shouldn't it be A+B?

[hypermorphism]

Why does the equation have C=A*B? Shouldn't it be A+B?
The sum of two 3-dimensional cartesian vectors A + B is already defined to be
(A_x + B_x)\hat{x} + (A_y + B_y)\hat{y} + (A_z + B_z)\hat{z}
The definition you have of the dot product is quite different from the addition of two vectors, and hence has a different symbol. :)

[UrbanXrisis]

how would I show in vector notaton that displacement described in problis given by s=1.08y[m]

What exactly is vector notation? And how is distance a vector? What exactly is the question asking for?

What is a dot product?

[hypermorphism]

how would I show in vector notaton that displacement described in problis given by s=1.08y[m]

What exactly is vector notation? And how is distance a vector? What exactly is the question asking for?

We generally don't use the word "distance" to describe a vector; it is usually reserved for the length of a displacement or position vector, which is a scalar. There are two different ways of writing vector notation, the most common being just an ordered list (x,y,z). Another method is to write out the identifying unit vectors for each coordinate as well, which we have used in defining the dot product and addition.
Each method uses components. A vector in Euclidean geometry can be described as in your first post by its angle or angles (in higher dimensions) from the axes and its length, as the component notation can then be easily derived from the Pythagorean theorem (or in different geometries, by the dot product).
In the case that your vector is two-dimensional and it is described as being at some angle \theta from the positive x-axis and having a length h, you can just use the right-triangle the vector forms with the axes to get the x-component as h*\cos\theta and the y-component as h*\sin\theta. From this, it seems the component notation in your first post is switched around.
What is a dot product?
The dot product is an invariant that allows one to get the projection (a scalar) of one vector onto another. This makes it useful to describe scalars such as Work, defined as the total magnitude of force exerted along the displacement vector of the object being worked upon, etc.
For example, you have a particle (a point) in 2-d Euclidean space and it experiences a force F, but moves along some displacement D (not necessarily parallel to F). If we want to know the work done on the particle by F, we want only the total force exerted parallel to the displacement of the particle, because the rest of the force acted orthogonally to the particle, and thus did not contribute to the displacement. Thus we want
|\|\vec{F}\|\cos\theta * \|D\|
(the magnitude of the force acting parallel to the displacement scaled by the magnitude of the displacement. The \theta comes from the right-triangle formed by the two vectors).