Fixed your LaTeX. The closing tag should be [ /tex], not [ \tex] (without the spaces). Also, you don't need the $ characters.Lightf said:Homework Statement
Show that for every quaternion z we have:
[tex]
\overline{z} = \frac{1}{2}(-z-izi - jzj - kzk)
[/tex]
That is the question, I just don't know how to begin and the " izi - jzj - kzk " confuses me. I need help on how to start this. Thanks a lot :D
A quaternion is a mathematical concept that extends the complex numbers and allows for the representation of three-dimensional rotations. It is represented by four components: a real part and three imaginary parts.
A quaternion is different from a complex number in that it has four components instead of two. The imaginary parts in a quaternion represent rotations in three-dimensional space, while in a complex number they represent rotations in two-dimensional space.
Showing that for every quaternion z we have means proving a statement that holds true for all possible quaternions. In this case, it refers to proving a property or equation that applies to any quaternion, regardless of its specific values for the four components.
One example of a quaternion is z = 2 + 3i + 5j + 7k, where i, j, and k are the imaginary units. To apply the statement "for every quaternion z we have", we can substitute this specific quaternion into an equation or property and show that it holds true. For example, we can show that for every quaternion z, the conjugate of z is equal to the product of z and its inverse.
Showing that a statement applies to every quaternion is important because it demonstrates the universality and validity of the statement. It allows us to apply the statement to any quaternion we encounter, without having to check for exceptions or special cases. This is crucial in developing mathematical theories and models that can be universally applied and relied upon in various fields of science and engineering.