Understanding Vector Multiplication: Examples and Differences Explained

[60051]

Not sure if vectors are pre or post calculus

Homework Statement

So we had a few examples of vector multiplication in class and I don't understand something.

The following vectors were multiplied:

[5 2 1] by

3
-4
7

...and the answer was 14 [(5*3) + 2*(-4) + (1*7)].

But in another example,

2
1

was multiplied by [-1 3]

...and the answer was

-2...6
-1...3

So why was the answer in the first example a single numerical answer while the answer to the second one was in matrix form?

 

[jegues]

It makes no difference, perhaps your prof was trying to get the point across that vectors can be represented in matrix form as well?

EDIT: After looking over the question and putting a little more thought into it,

The first part it looks as though you're preforming the dot product between two vectors.

The second part looks like the matrix multiplication of a row vector and a column vector to generate a 2x2 matrix.

 

[pbandjay]

They are both still examples of matrix multiplication. The first example is a 1x3 matrix multiplied by a 3x1 matrix, and the result is then a 1x1 matrix. The second example is a 2x1 matrix multiplied by a 1x2 matrix, and the result is a 2x2 matrix.

 

[HallsofIvy]

With "matrix multiplication". Think of matrix multiplication of a "dot product" of row and column. To find the number in "column i, row j" of the result, take the dot product of "row j" in the first matrix with "column i" in the second.

Specifically,
[tex]\begin{bmatrix}5 & 2 & 1\end{bmatrix}\begin{bmatrix}2 \\ -4 \\ 7\end{bmatrix}= 14[/itex]
The dot product of the single row in the first matrix with the single column in the second. A "row matrix" times a column matrix is a number.

[tex]\begin{bmatrix}2 \\ 1\end{bmatrix}\begin{bmatrix} -1 & 3\end{bmatrix}= \begin{bmatrix}-2 & 6 \\ -1 & 3\end{bmatrix}[/itex].
"Row 1 is just the number 2 and "column 1" is just the number -1. Their product is -2. "Column 2" is just the number 3 so the product of "row 1 and column 2" is (2)(3)= 6. "Row 2" is just 1. The product of "row 2 and column 1" is (1)(-1)= -1. The product of "row 2" and "column 2" is (1)(3).

 

[tiny-tim]

Hi 60051!

So we had a few examples of vector multiplication …

So why was the answer in the first example a single numerical answer while the answer to the second one was in matrix form?

They are both still examples of matrix multiplication. The first example is a 1x3 matrix multiplied by a 3x1 matrix, and the result is then a 1x1 matrix. The second example is a 2x1 matrix multiplied by a 1x2 matrix, and the result is a 2x2 matrix.

As pbandjay says, they are matrix multiplication.

Vector multiplication doesn't really exist.

For multiplication, you need everything to be in a multiplication space (my terminology) … both the input and the output.

The product of two matrices is always another matrix (possibly a different size and shape).

The dot product of two vectors isn't a vector, so the dot product isn't multiplication unless you regard everything (both vectors and scalars) as matrices.

So think of this as matrix multiplication, and don't use the phrase "vector multiplication". ​

 

[BigFairy]

You are correct.

 

[tiny-tim]

You are correct.

oooh … do i get three wishes?

are you a fairy operator? do you do transformations?