Vectors, Geometric vectors, Space, Vector operations, Scalar multiplication, Addition, Multiplication, Commutativity, Associativity, Dot product, Dyadic product.
Matrices with a single column are called vectors.
Vectors are indicated by underlined lower case letters.
Here are two vectors:
Vector is a vector, and vector is a vector, but it is redundant to say the part, as we already know they have only one column - that's why we call them vectors.
So, it is plenty enough if we just say how many numbers are in the vector. These numbers are called the coordinates of the vector.
It is comforting to know that what we recognize as a vector in geometry,
and what we defined as a vector just now, correspond to each other.
If we take three lines in space so that
they are perpendicular to each other, and then
we put a scale on them, then geometric vectors
can be represented by number triplets.
So, when we talk about vectors, we could be thinking about
matrices and also geometric forms.
Let's see what kind of operations we can do with vectors.
1. SCALAR MULTIPLICATION
Dot product (also called scalar product or inner product): Dyadic product (also called tensor product or outer product):
From these two types of products the dot product
will be much more useful for us, so at this point
we say good bye to dyadic products.
As to the dot product, let's introduce
a simple notation.
This way we can save a few *.
But let's see what else we can do with dot products.