Conceptual definitions of vector topics

[NJJ289]

(Is this thread in the right place?)

A few questions on vectors:

1. I was wondering if anyone could explain in conceptual terms what the dot product and cross products represent.

(I understand how these are calculated, but why are they important?)

2. Also, would it be accurate to describe a (m<4 x n<4) matrix as either a vector, point, line or plane? If not, what does a matrix represent in geometric terms?

3. What operation is used for vector multiplication that is unspecified (does not use a dot or a cross)?

4. Is there any difference in the top hat notation and the arrow notation for indicating vectors?Thanks for the assistance!

 

[HallsofIvy]

(Is this thread in the right place?)

A few questions on vectors:

1. I was wondering if anyone could explain in conceptual terms what the dot product and cross products represent.
(I understand how these are calculated, but why are they important?)

$\vec{u}\cdot\vec{v}= |\vec{u}||\vec{v}|cos(\theta)$ so that the dot product can be used to define the angle between two vectors in an abstract "vector space". Of course two vectors are perpendicular if and only if their dot product is 0.

Of course, the cross product of two vectors is always perpendicular to both so it is a good way of finding "normal" vectors. Also, if you think of vectors $\vec{u}$ and $\vec{v}$ as giving to non-parallel sides of a parallelogram, then the area of the parallelogram is $\vec{u}\times\vec{v}$.

2. Also, would it be accurate to describe a (m<4 x n<4) matrix as either a vector, point, line or plane? If not, what does a matrix represent in geometric terms?

None of those. An m by n matrix is a linear function that maps an m dimensional vector to a n dimensional vector.

3. What operation is used for vector multiplication that is unspecified (does not use a dot or a cross)?

None. $\vec{u}\vec{v}$ simply is meaningless. That is just bad notation. If you saw it in a book or elsewhere, it might be some especially defined meaning. Can you give an example?

4. Is there any difference in the top hat notation and the arrow notation for indicating vectors?

Not really, although some texts specifically use the notation $\hat{v}$ to refer to the basis vectors. That is to write $\vec{v}= a\hat{i}+ b\hat{j}+ c\hat{k}$

Thanks for the assistance!

 

[Stephen Tashi]

(Is this thread in the right place?)

It will be OK here. I think you could have posted it in the Mathematics section since this is not a specific "homework type" problem.

A few questions on vectors:

1. I was wondering if anyone could explain in conceptual terms what the dot product and cross products represent.

(I understand how these are calculated, but why are they important?)

One way to think of these is to think of their applications to physics.

We say that "Work equals force time distance", but that is only in the case where the force vector points in the direction of the distance. How do you calculate the work done when the force points differently? For a constant force vector, we are supposed to take the component of that vector that points in the direction of motion and multiply the length of that component times the distance moved. You can do that by trigonometry. But you can also do it by representing the vectors in cartesian coordinates and taking their dot product. To show you get the same answer is an interesting exercise in algebra and trigonometry.

To compute if a see-saw balances we equate torques. Torque about a pivot point = (force perpendicular to the lever arm) times length of the lever arm. What do you do when the force isn't perpendicular to the lever arm? You take the magnitude of the component of the force that is perpendicular to the lever arm and multiply it by the lever arm. That can be done by trigonometry. It can also be done by expressing the force and the lever arm as vectors in cartesian coordinates and taking their cross product. Since you get a vector as an answer, this also adds the information about the plane of the torque and the axis about which it "wants to turn". Showing the two different methods produce the same answer is an exercise is algebra and trigonometry.

2. Also, would it be accurate to describe a (m<4 x n<4) matrix as either a vector, point, line or plane? If not, what does a matrix represent in geometric terms?

Technically you can say that matix is a vector or even a point by pretending all its entries are listed in some linear order. However, this is not the most useful way to think about matrices and most mathematical articles don't want you to do that.

One way to think about a matrix is that it represents a change of coordinates. Another almost equivalent way is to think about it as a formula that represents certain transformations ("linear transformations", to be precise) that change the shape of an object. For example, multiplication of a matrix times a column vector produces a different column vector.

$$\begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix} \begin{pmatrix}3 \\ 4 \end{pmatrix} = \begin{pmatrix} 11 \\ 2 \end{pmatrix}$$

You can think about that in two different ways: You can think of it as keeping the vector represented by the column vector the same and changing the coordinate system to a new one so that the coordinates of the vector become (11,2). Or you can think of this as keeping the the same coordinate system and moving the vector to a new vector. Many mathematical presentations take the former view. One that emphasizes the latter view is the thin, easy book Matrices and Transformations by Anthony J. Pettofrezzo.

Linear transformations of objects (accomplished by transforming the vectors that are the vertices) include, as special cases, rotation, reflection, similarity (i.e. enlarging or reducing by a constant factor) etc. You can even represent the translation of an object this way if you adopt what are called "homgeneous" or "projective coordinates" instead of the usual cartesian coordinates.

3. What operation is used for vector multiplication that is unspecified (does not use a dot or a cross)?

You'd have to read the particular article that uses this notation carefully to find out. There isn't a widely accepted agreement on what that notation would mean.

4. Is there any difference in the top hat notation and the arrow notation for indicating vectors?

The top hat often indicates a vector that has length 1, but you still should read the article that uses this notation carefully to make sure what it means.