Complex Numbers and Vector Multiplication

In summary, the product of two complex numbers, which can be used to represent vectors, is still a complex number. However, vectors cannot be multiplied in the same way as complex numbers, as this would violate the symmetries of the vector space. Instead, there are other operations such as the dot product and the cross product, which result in a scalar and a pseudovector, respectively. To understand these concepts further, it is helpful to research topics such as vector spaces and wedge products.
  • #1
iampaul
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I have read from my algebra book that the product of two complex numbers is still a complex number: (a+bi)(c+di)= (ac-db)+(bc +ad)i
I was thinking that since complex numbers can be used to represent vectors, the product of two vectors should still be a vector. But I have also read from my physics book that there are two ways to multiply vectors, which are the dot product and the cross product.The dot product is the product of two parallel vectors and results into a scalar, while the cross product is the product of two perpendicular vectors. Why should these products be defined differently? Using complex no.s, two vectors whether parallel or not should still yield a product which is still a vector. Are these dot and cross products different from ordinary complex number multiplication? If so, when do we use the ordinary complex no. or vector multiplication? Am i missing anything? What math topics should i read? Please reply, I'm really getting confused.

Any help will be greatly appreciated!
 
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  • #2
hi iampaul! :smile:
iampaul said:
I have read from my algebra book that the product of two complex numbers is still a complex number: (a+bi)(c+di)= (ac-db)+(bc +ad)i
I was thinking that since complex numbers can be used to represent vectors, the product of two vectors should still be a vector.

vectors exist in a vector space

a vector space is defined as having addition of vectors, and multiplication of a vector by a by a scalar (to make a vector)

we can also define an inner product (essentially, combining two vectors to make a scalar), a wedge product (essentially, combining two vectors to make a "2-form", which is something like a tensor), and a dual (eg in 3D, the dual of a vector is the 2-form "perpendicular" to it)

we cannot define a combination of two vectors to make another vector (other than addition, of course) unless we abandon the symmetries of the space

for example, in your 2D case, we could define a "complex vector product" to be the result of multiplying the corresponding complex numbers …

but then y times y would be -x (because i times i = -1), but x times x would be x

this is unsymmetric: rotating the whole space 90° should leave everything looking the same, but it makes your very simple "complex vector product" look completely different! :smile:
But I have also read from my physics book that there are two ways to multiply vectors, which are the dot product and the cross product.The dot product is the product of two parallel vectors and results into a scalar, while the cross product is the product of two perpendicular vectors.

No, the dot product is a product of any two vectors (and results in a scalar), while the https://www.physicsforums.com/library.php?do=view_item&itemid=85" is the product of any two vectors, and results in a pseudovector (which teachers usually wrongly tell you is another vector, since after all it does look exactly like one :rolleyes:).

The dot product is an inner product. The cross product is the dual of a wedge product.
What math topics should i read?

google "vector space" and "wedge product" :wink:
 
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Related to Complex Numbers and Vector Multiplication

1. What are complex numbers and vector multiplication?

Complex numbers are numbers that are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Vector multiplication is a mathematical operation where two vectors are multiplied together to produce a new vector.

2. How are complex numbers and vector multiplication related?

Complex numbers can be represented as vectors in a two-dimensional plane, with the real part represented on the x-axis and the imaginary part on the y-axis. Vector multiplication of two complex numbers involves multiplying their magnitudes and adding their angles.

3. What is the difference between scalar multiplication and vector multiplication?

Scalar multiplication involves multiplying a vector by a single number, resulting in a new vector with a different magnitude. Vector multiplication involves multiplying two vectors, resulting in a new vector with a different direction and magnitude.

4. What are some real-world applications of complex numbers and vector multiplication?

Complex numbers and vector multiplication are used in many fields, including engineering, physics, and computer graphics. They are used to model and analyze alternating current circuits, quantum mechanics, and 3D animations, among other things.

5. Can vector multiplication be performed on any type of vector?

No, vector multiplication can only be performed on vectors of the same dimension. For example, a 2-dimensional vector can only be multiplied by another 2-dimensional vector, not a 3-dimensional vector. Additionally, the order in which the vectors are multiplied matters and can affect the result.

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