Contents of this Calculus 3 tutorial:

Vectors, Geometric vectors, Space, Vector operations, Scalar multiplication, Addition, Multiplication, Commutativity, Associativity, Dot product, Dyadic product.

Text of slideshow

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.

VECTOR OPERATIONS

1. SCALAR MULTIPLICATION

Example:

2. ADDITION

Example:

PROPERTIES:

Commutative:

Associative:

3. MULTIPLICATION

Dot product (also called scalar product or inner product): Dyadic product (also called tensor product or outer product):

PROPERTIES:

Commutative:

Not associative:

Example:

and

and

Dot product:

Dyadic product:

PROPERTIES:

Not commutative

Not associative

Example:

and

Dyadic 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.

 

Vector operations

03

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