Thread: Line through two points in field C X C

This should be simple but is giving me fits. Given two points in field C X C (C is field of complex numbers), how do you determine the equation of the line? For example, two points are (2,5) and (i,10).

Originally Posted by Pointer

This should be simple but is giving me fits. Given two points in field C X C (C is field of complex numbers), how do you determine the equation of the line? For example, two points are (2,5) and (i,10).

Try thinking of $\displaystyle \mathbb{C}^2 = \mathbb{C} \times \mathbb{C}$ as $\displaystyle \mathbb{R}^4$. You know what lines here look like, and elements in $\displaystyle \mathbb{C}^2$ look like $\displaystyle (a+ib, c+id)$ but can be thought of as $\displaystyle (a, b, c, d)$.

Or, just do it as you would in $\displaystyle R^2$: y= mx+ b but now m, b, x, and y can be complex numbers.

Since (2, 5), x= 2 and y= 5 must satisfy the equation. 5= 2m+ b. Since (i, 10) is on the line, x= i and y= 10 must satisfy the equation. 10= im+ b.

subtracting the first equation from the first eliminates b and gives 5= (i- 2)m. Then $\displaystyle m= \frac{5}{i- 2}= \frac{5}{i- 2}\frac{-i- 2}{-i- 2}$$\displaystyle = \frac{10- 5i}{1+ 4}= 2- i$. Now you can use either of those equations to solve for b.