Is the Result of a Cross Product in R^3 a Vector in R^3?

d_b
Messages
36
Reaction score
0
Hi everyone...

I just wanted to know if you compute a cross product of two vectors in R^3, do you get a vector in R^3 or an actual value(say both vectors have actually values)...

Another question. I did this in class but I wasn't sure how it would work. Let say I have a metrix 2x2. how do i check to see if a metrix {[a b(first row), c 1(bottom row)/ a,b,c in R], is a vector space?

now If I'm not wrong it isn't a vector space because if fail the second axiom (addition between 2 vectors) but my TA said it is a vector space...so could you guys show me how it is a vector space??

thank you... :D
 
Physics news on Phys.org
d_b said:
I just wanted to know if you compute a cross product of two vectors in R^3, do you get a vector in R^3 or an actual value(say both vectors have actually values)...
You get a vector which is perpendicular to both. Cross product of 2 vectors yield another vector.

Another question. I did this in class but I wasn't sure how it would work. Let say I have a metrix 2x2. how do i check to see if a metrix {[a b(first row), c 1(bottom row)/ a,b,c in R], is a vector space?

now If I'm not wrong it isn't a vector space because if fail the second axiom (addition between 2 vectors) but my TA said it is a vector space...so could you guys show me how it is a vector space??
I don't see how it's a vector space either. As you said, adding two of these matrices (in the standard way of matrix addition since you didn't specify the vector additive operation) yield a matrix whose bottom right entry is 2, which clearly doesn't belong to the set.
 
d_b said:
Hi everyone...

I just wanted to know if you compute a cross product of two vectors in R^3, do you get a vector in R^3 or an actual value(say both vectors have actually values)...
What do you mean by "an actual value". The cross product of two vectors in R3 is a vector in R3 which, in my opinion, is an "actual value"! If, by "actual value" you mean a number (scalar) then the answer is no, you get a vector in R3.

Perhaps, when you say "both vectors have actual values" you mean the "absolute value" or length of the vector. The formula you often see, "|u||v|sin(\theta)", is not for the cross product but only the length of the cross product. The cross product of two vectors, u x v, is a vector in R3 having that number as length. The direction is perpendicular to both, using the "right hand rule", so that u x v is the opposite direction to v x u.

Another question. I did this in class but I wasn't sure how it would work. Let say I have a metrix 2x2. how do i check to see if a metrix {[a b(first row), c 1(bottom row)/ a,b,c in R], is a vector space?

now If I'm not wrong it isn't a vector space because if fail the second axiom (addition between 2 vectors) but my TA said it is a vector space...so could you guys show me how it is a vector space??

thank you... :D
"matrix", not "metrix".

You are asking if the set of all matrices of the form
\left[\begin{array}{cc}a & c \\ b & 1\end{array}\right]
form a vector space with the usual operations?

No, it doesn't because, as you say, the sum of two such matrices is not of that form. However, if a\ne bc it does form a vector space with matrix multiplication as operation.
 
hm...guess i have to go talk to my ta then. For the cross product, yes I did mean to say the absolute value. Thanks for clearing that up for me... :D
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I About the definition of vector space of infinite dimension
Replies
18
Views
3K
I Terminologies used to describe tensor product of vector spaces
Replies
7
Views
3K
I Vector subspace and basis vectors in the context of data science
Replies
6
Views
3K
B What is the cross product of two vectors?
Replies
32
Views
4K
I Dimension and solution to matrix-vector product
Replies
4
Views
3K
A Reason out the cross product (for the moment): a skew symmetric form
Replies
14
Views
3K
A What is a free product of groups or vector space?
Replies
2
Views
2K
I Is tensor product the same as dyadic product of two vectors?
Replies
4
Views
3K
I Visualization of complex vectors and dot product
Replies
14
Views
4K
B Confused about dot product multiplication
Replies
33
Views
4K

Hot Threads

Recent Insights

Back
Top