Understanding Column Vectors: A Physical Interpretation

[Saladsamurai]

Column Vectors?

Alrighty. I have a silly question about column Vectors. I first learned about the concept of a vector in Physics, i.e. position, velocity, acceleration, etc. so I easily understand the premise of a row vector. For example the matrix \left[\begin{array}{ccc}1 & 2 & 3\end{array}\right] I can easily see possibly represents a position vector x+2y+3z which is a point in three-dimensional space i.e. in the x,y,z planes.

Okay... great. Now the column vector. What is it? Looking at the matrix
\left[\begin{array}{c}1\\2\\3\end{array}\right],
I can't as easily see what is represents? That is, why is it a vector?

Can someone give me a physical interpretation of the column vector if possible?

Thanks

 

[Dick]

Technically, a column vector is an element of the dual space to a row vector. If you don't know what that means, I wouldn't stress about it at this point. If you are defining the dot product as x.y=x(y^T) then it's just a trick to define the dot product in terms of matrix multiplication. They are both just 'vectors'. But if you look at how vectors transform under coordinates changes then not all 'vectors' are created equal. Tangent vectors to curves transform one way, gradient vectors transform another way. It makes life easier to call one 'row' and the other 'column'. That's all.

 

[jimvoit]

Linear algebra is a pretty big enterprise, but here's one answer that's even sort of silly..
If you are explaining to someone how to multiply two 3X3 matrices, and your left index finger is moving along the rows of the first matrix, what will your right index finger be doing?
(sorry)

 

[Mute]

Generally in physics you don't really distinguish between column vectors or rows vectors. It's just whichever notation you prefer: do you like writing your vectors as rows or columns? They mean the same thing, you just have to be consistent with your choice of the way to write it. Usually people use column vectors so that when they use matrix notation it looks like \mathbf{y} = \mathsf{A}\mathbf{x}, where y and x are column vectors and A is a matrix.

 

[Defennder]

A column vector is just a nx1 matrix. A row vector is just a 1xn matrix. Your question is like asking for the physical interpretation of say, nabla in vector calculus. It's just a convention of notation which is given its physical significance in the context whereby it's used. You can use the column vector say for example to represent a vector in rectangular basis vectors:
\left(\begin{array}{c}1\\1\\1\end{array}\right)= i + j + k