Confusion regarding dot product of vectors(row matrices)

[mdnazmulh]

I’ve got a confusion. We know a 1x3 row matrix is a 3-vector i.e.
x= [ a b c]
Matrix x can be written in vector notation like x= a i + b j + c k
where i, j, k are unit vectors along x,y & z axes.
For dot product of
x.x = a² + b² + c² when x= a i + b j + c k

But according to the matrix multiplication rule, multiplication of two matrices is possible only when column of 1st matrix = row of the 2nd matrix.
So x.x = [ a b c] [ a b c] is not possible
My questions are :
(1) Both x= [ a b c] and x= a i + b j + c k are same vector.
Then why this discrepancy happens?
(2) Does really x.x exist when x = [ a b c]? Can we approach in any other way to define x.x when x = [ a b c] ?
I’m novice at linear algebra. So it would be helpful for me if you can explain elaborately. I’m really at a loss about that confusion.

 

[tiny-tim]

For dot product of
x.x = a² + b² + c² when x= a i + b j + c k

But according to the matrix multiplication rule, multiplication of two matrices is possible only when column of 1st matrix = row of the 2nd matrix.

Hi mdnazmulh!

Technically, the product for "matrix" vectors is an inner product, and one of the vectors must be a transpose vector (written as a column vector instead of a row vector, or abbreviated x^T).

So the inner product is xx^T.

For details, you could see http://en.wikipedia.org/wiki/Inner_product_space" …

but I wouldn't bother until your professor deals with it.

 

[mdnazmulh]

Thanks for your reply. Actually in the book of introductory linear algebra by Bernard Kolman there is a question that if x is an n-vector then is it possible that x.x can have negative value? And part (b) of the question says that if x.x=0 , what is x=0?
Now I understand the author placed those questions in the exercise just to check our conception. Answer MUST BE that x.x is no way possible.
Thank u again

 

[tiny-tim]

… Now I understand the author placed those questions in the exercise just to check our conception. Answer MUST BE that x.x is no way possible.

No, that's not what I meant.

For a vector that's just a vector, x.x is correct.

Only if a vector is considered as a matrix (which I called '"matrix" vectors'), does the product have to be xx^T …

that's what you originally asked about.