Cross product and vector space

In summary, the conversation discusses the computation of a cross product of two vectors in R^3, and whether it results in a vector or a scalar value. It also touches on the topic of a specific matrix being a vector space and the confusion surrounding it. The summary concludes that the cross product yields a vector in R^3 and the mentioned matrix is not a vector space due to failing the second axiom of vector addition.
  • #1
d_b
36
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
 
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  • #2
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.
 
  • #3
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, "[itex]|u||v|sin(\theta)[/itex]", 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
[tex]\left[\begin{array}{cc}a & c \\ b & 1\end{array}\right][/tex]
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 [itex]a\ne bc[/itex] it does form a vector space with matrix multiplication as operation.
 
  • #4
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
 

Related to Cross product and vector space

1. What is a cross product?

A cross product, also known as a vector product, is a mathematical operation that takes two vectors as inputs and produces a new vector that is perpendicular to both of the original vectors.

2. How is the cross product calculated?

The cross product of two vectors, a and b, can be calculated using the following formula: a x b = |a| |b| sin(θ) n, where |a| and |b| are the magnitudes of the vectors, θ is the angle between the two vectors, and n is the unit vector perpendicular to both a and b.

3. What is the significance of the cross product in vector spaces?

The cross product is important in vector spaces as it allows us to determine the direction of a vector that is perpendicular to two other vectors. It is also useful for calculating the area of a parallelogram or determining the orientation of a plane.

4. Can the cross product be applied to vectors in any dimension?

No, the cross product can only be applied to vectors in three-dimensional space. In two-dimensional space, the equivalent operation is known as the scalar or dot product.

5. What is the difference between the cross product and the dot product?

The cross product produces a vector as its result, while the dot product produces a scalar. The cross product also takes two vectors as inputs and produces a vector that is perpendicular to both, while the dot product takes two vectors as inputs and produces a scalar that represents the projection of one vector onto the other.

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