Understanding Vector Multiplication

[Hopjopper]

Hi i am trying to understand this thoroughly

Basically i am trying to understand vector multiplication, i don't know if it is the cross product or the dot product i am thinking of

Okay so here is the question and what is confusing me in the answer

So if we have two vectors and we multiply them

a.b this in my mind as i understand it means this(in 2d):

(a_x + a_y ) * (b_x +b_y) = a_x * b_x + a_x * b_y + a_y * b_x + a_y * b_y

now i don't understand why the dot product misses these in the centre? and goes straight to only a_x * b_x + a_y * b_y (or the other way a.b.cos(theta))

what is it exactly that is being multiplied here? and furthermore what exactly is being found here?

if i wanted to move a vector from position a -> position b , by using the dot product am i finding that space in between? (i.e the vector which is required to be added to a to transform vector a to b?)

its really confusing me all of this?

 

[PeroK]

Your confusion is quite fundamental. You probably need to find a good introductory text on vectors and vector operations and study it.

The dot product shouldn't be hard to understand. It has an algebraic and a geometric significance and has lots of applications in physics.

A good textbook will explain all this.

 

[Hopjopper]

Your confusion is quite fundamental. You probably need to find a good introductory text on vectors and vector operations and study it.

The dot product shouldn't be hard to understand. It has an algebraic and a geometric significance and has lots of applications in physics.

A good textbook will explain all this.

can you just explain this to me if you don't mind

what happens to a_x * b_y + a_y * b_x

 

[PeroK]

Nothing happens to them. These terms are simply not part of the dot product.

 

[A.T.]

So if we have two vectors and we multiply them

a.b this in my mind as i understand it means this(in 2d):

(a_x + a_y ) * (b_x +b_y)

Well, of course you can define new types of vector products, based on what jumps into your mind. What you wrote above is the product of Manhattan norms:
http://en.wikipedia.org/wiki/Norm_(mathematics)#Taxicab_norm_or_Manhattan_norm

 

[Hopjopper]

okay i have understood this and sorted it out

this is what i was looking for

i did confuse the dot product with vector addition

so if i have a vector


/|
/ |
/ | 7
/ |
5

and i wanted to move this point to say
/|
/ |
/ |8
/ |
3

i would need to add the vector

/|
/ | 1
/ |
-2

and dot product is actually only multiplying a vector which has nothing to do with the components but rather with the complete vector magnitude itself

such as a . b would be (if the angle between them is 15 degrees/radians)

a.b cos(15) because you would be getting the component of b which is in line with vector a so that they are both in the same direction and simply multiply them as if they are another scalar * vector multiplication

 

[Hopjopper]

Well, of course you can define new types of vector products, based on what jumps into your mind. What you wrote above is the product of Manhattan norms:
http://en.wikipedia.org/wiki/Norm_(mathematics)#Taxicab_norm_or_Manhattan_norm

now that i have understood vectors, they are an amazing tool for cognitive thought! such powerful mathematical tools!

 

[jtbell]

(a_x + a_y ) * (b_x +b_y) = a_x * b_x + a_x * b_y + a_y * b_x + a_y * b_y

Your vectors are incomplete because they don't include the unit vectors: $$\vec A = a_x \hat x + a_y \hat y \\ \vec B = b_x \hat x + b_y \hat y$$ The product is $$ \vec A \cdot \vec B = (a_x \hat x + a_y \hat y) \cdot (b_x \hat x + b_y \hat y) \\ \vec A \cdot \vec B = a_x b_x (\hat x \cdot \hat x) + a_x b_y (\hat x \cdot \hat y) + \cdots$$ I'll let you fill in the rest. Some of the dot products of unit vectors equal zero, and some equal 1.