Confusion regarding dot product of vectors(row matrices)

In summary, the conversation discusses the confusion regarding the dot product of a 1x3 row matrix, which can also be written in vector notation. The discrepancy arises from the fact that for matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Therefore, the dot product of x.x, where x is a 1x3 row matrix, is not possible. The conversation also touches on the concept of inner product and its relation to matrix vectors. The conversation concludes by mentioning a question from a linear algebra book about the possibility of x.x having a negative value and the clarification that x.x is not possible for a regular vector, but only for a "matrix" vector.
  • #1
mdnazmulh
51
0
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.
 
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  • #2
mdnazmulh said:
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! :smile:

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 xT).

So the inner product is xxT.

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

but I wouldn't bother until your professor deals with it. :smile:
 
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  • #3
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
 
  • #4
mdnazmulh said:
… 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 xxT …

that's what you originally asked about. :wink:
 

1. What is the dot product of two vectors?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors as input and returns a single scalar value. It is calculated by multiplying the corresponding elements of the two vectors and then summing up the products.

2. How is the dot product represented?

The dot product can be represented in two ways: algebraically as a numerical value or geometrically as the cosine of the angle between the two vectors. Algebraically, the dot product is denoted by a dot (·) or by enclosing the vectors in brackets and placing a dot between them. Geometrically, it is represented as the length of one vector multiplied by the length of the projection of the other vector onto the first vector.

3. What is the purpose of the dot product?

The dot product is used to calculate the angle between two vectors, determine if two vectors are perpendicular, and find the projection of one vector onto another. It also has many applications in physics and engineering, such as calculating work, power, and torque.

4. Can the dot product of two row matrices be calculated?

Yes, the dot product can be calculated for any two vectors, regardless of their representation. In the case of row matrices, the dot product is calculated by multiplying the corresponding elements of each row and then summing up the products.

5. What is the difference between the dot product and the cross product?

The dot product and the cross product are two different mathematical operations involving vectors. The dot product results in a scalar value, while the cross product results in a vector. Additionally, the dot product measures how much two vectors are aligned, while the cross product measures the area of the parallelogram formed by the two vectors.

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