The logic behind the cross product with units

In summary, the cross product of two vectors in R^3 represents the area of the parallelogram defined by the two vectors. If the vectors have units of meters, the cross product will have units of square meters. It can also be thought of as representing an area between the two vectors or as a vector normal to a surface. The cross product, along with the dot product, can be used to find derivatives of surfaces.
  • #1
ManDay
159
1
The cross product equals to the area of the parallelogram defined by the two vectors (at least in R^3). So if working on vectors which units

v_1 = (1,2,3)m
v_2 = (3,4,5)m

it correctly returns the according area. However, if used to get a vector perpendicular to each of the vectors the resulting vectors has the unit m^2.

How does this fit together? Whats the term for the "dimension" of this vector anyway, since it's actual dimension is 3 (am I mistaken here?) as in the field of R^3.

Thanks for your help, PF
 
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  • #2
ManDay said:
The cross product equals to the area of the parallelogram defined by the two vectors (at least in R^3). So if working on vectors which units

v_1 = (1,2,3)m
v_2 = (3,4,5)m

it correctly returns the according area. However, if used to get a vector perpendicular to each of the vectors the resulting vectors has the unit m^2.

How does this fit together? Whats the term for the "dimension" of this vector anyway, since it's actual dimension is 3 (am I mistaken here?) as in the field of R^3.

Thanks for your help, PF
I'm not clear what your question is. Yes, v_1, v_2, and their cross product all have dimension 3. But I don't know what you mean by "the term for the dimension".

If the "m" on v_1 and v_2 means "meters" and denotes that the components are in meters, yes, their cross product will have units of m^2, an area. General vectors, of course, either don't have units, or if in a specific application, units appropriate to that application. In that case the cross product of two vectors will have "square units" since it is a multiplication- just as the dot product of two such vectors will have "square units".

Yes, you can think of the cross product of two vectors as representing an area: here the area "between" the two vectors. In fact, if you are given a surface represented by a vector equation, [itex]\vec{r}= f(u,v)\vec{i}+ g(u,v)\vec{j}+ h(u,v)\vec{k}[/itex] with the two parameters u and v, then the derivatives with respect to u and v, [itex]\vec{r}_u= f_u(u,v)\vec{i}+ g_u(u,v)\vec{j}+ h_u(u,v)\vec{k}[/itex] and [itex]\vec{r}_v= f_v(u,v)\vec{i}+ g_v(u,v)\vec{j}+ h_v(u,v)\vec{k}[/itex] are vectors tangent to the surface and their cross product is a vector normal to the surface and its length is the "differential of surface area".
 

Related to The logic behind the cross product with units

What is the cross product and why is it important in science?

The cross product is a mathematical operation that takes two vectors and returns a third vector that is perpendicular to both of the original vectors. It is important in science because it allows for the calculation of quantities such as torque, magnetic force, and angular momentum.

What are the units of the cross product and how are they calculated?

The units of the cross product are a combination of the units of the two original vectors. For example, if the two vectors have units of meters and seconds, then the cross product will have units of meters per second. The specific calculation of units depends on the specific application and the quantities being multiplied.

Why does the cross product use the right-hand rule?

The right-hand rule is used to determine the direction of the resulting vector in the cross product. This rule states that if the fingers of the right hand curl in the direction of the first vector and then curl towards the second vector, the resulting vector will point in the direction of the thumb. This convention is used to maintain consistency and clarity in the direction of the resulting vector.

How is the cross product related to the dot product?

The cross product and the dot product are two different operations that can be performed on vectors. The dot product results in a scalar quantity, while the cross product results in a vector quantity. Additionally, the dot product measures the similarity or projection of two vectors, while the cross product measures the perpendicularity or rotational relationship between two vectors.

What are some real-world applications of the cross product?

The cross product has many applications in physics, engineering, and other fields of science. Some examples include calculating the torque on a spinning object, determining the direction of magnetic force on a charged particle, and calculating the angular momentum of a rotating object. It is also used in vector calculus and computer graphics.

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