# Understanding Vector Multiplication

• Hopjopper
In summary: That's why the cross terms disappear.In summary, the conversation revolves around the confusion regarding vector multiplication, specifically the dot product and its components. The individual is seeking clarification on what exactly is being multiplied and what is being found in the process. They also mention their struggle in grasping the concept and request for an explanation. The expert suggests finding a good introductory text on vectors and their operations for a better understanding. The confusion is ultimately resolved by understanding the difference between dot product and vector addition, and the power of vectors as a mathematical tool.
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?

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.

PeroK said:
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

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

Last edited:
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 said:
(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.

## 1. What is vector multiplication?

Vector multiplication is a mathematical operation that combines two vectors to produce a new vector. There are two types of vector multiplication: dot product and cross product.

## 2. What is the purpose of vector multiplication?

The purpose of vector multiplication is to determine the relationship between two vectors, such as their direction and magnitude. This is useful in many scientific and engineering fields, including physics and computer graphics.

## 3. How is vector multiplication performed?

The method for performing vector multiplication depends on whether it is a dot product or cross product. Dot product involves multiplying the corresponding components of the two vectors and summing them, whereas cross product involves using a specific formula to calculate the new vector.

## 4. What are some real-world applications of vector multiplication?

Vector multiplication is used in various fields, such as physics, engineering, and computer graphics. Some examples of real-world applications include calculating forces in a physics problem, determining the direction and speed of an object's motion, and creating 3D graphics in video games and movies.

## 5. Can vector multiplication be applied to more than two vectors?

Yes, vector multiplication can be applied to any number of vectors. In some cases, it may be necessary to perform multiple multiplication operations in order to determine the final vector.

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