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

[d_b]

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

 

[Defennder]

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.

 

[HallsofIvy]

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.

 

[d_b]

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