shen07 said:Hi guys, i am solving a problem but I am unable to figure out what is $$\hat{\jmath} \times r$$? Please help me
Sudharaka said:Hi shen07, :)
Obviously what is meant by $\hat{\jmath} \times r$ depends on the context. Can you post the full question please?
shen07 said:I just edited it
$\hat{\jmath} \times r$ is the cross product of $\hat{\jmath}$ and $r$, but you already know this. Are you asking how to compute it? Then see here. Are you asking what $\hat{\jmath}$ and $r$ are? It looks like $\hat{\imath}$, $\hat{\jmath}$ and $\hat{k}$ are unit coordinate vectors. I don't know what $r$ is. Or are you asking what the physical meaning of the term $\hat{\jmath} \times r$ is?shen07 said:A fluid motion has velocity $$\underline{u}=\sin{(at)}\hat{\imath}+\hat{\jmath} \times r +\cos{(at)}\hat{k}$$
I need to know what is $$\hat{\jmath} \times r$$ to find Vorticity and other things.
A cross product is a mathematical operation performed on two vectors in three-dimensional space that results in a new vector perpendicular to both of the original vectors. It is used in many areas of mathematics and science, including computer graphics and physics.
To compute a cross product, you need to take the two vectors and use a specific formula to calculate the resulting vector. The formula involves finding the determinant of a 3x3 matrix made up of the components of the original vectors. The resulting vector will have three components, representing the x, y, and z directions.
A cross product has many applications in mathematics and science. It is commonly used in computer graphics to calculate lighting and shading effects, as well as in physics to determine the torque on an object. It can also be used to find the area of a parallelogram or to determine if two vectors are parallel or perpendicular.
No, a cross product can only be computed in three dimensions. This is because the formula for computing a cross product involves a 3x3 matrix, and anything beyond three dimensions would require a larger matrix, which does not exist. However, there are similar operations that can be performed in higher dimensions, such as the wedge product.
One limitation of computing a cross product is that it can only be done on vectors in three-dimensional space. Additionally, the result of a cross product is always perpendicular to the original vectors, so it cannot be used to find the angle between two vectors. Also, the cross product is not commutative, meaning changing the order of the vectors will result in a different vector. Lastly, the cross product is only defined for real numbers, so it cannot be used with complex numbers.