Final answer:
The parametric equations of the line passing through points (1,4,2) and (3,5,0) are x = 1 + 2t, y = 4 + t, and z = 2 - 2t. The symmetric equations of the line are \(\frac{x-1}{2} = \frac{y-4}{1} = \frac{z-2}{-2}\).
Step-by-step explanation:
To find the parametric equations of the line passing through points (1,4,2) and (3,5,0), we first find the direction vector d by subtracting the coordinates of the first point from the second: d = (3-1, 5-4, 0-2) = (2,1,-2).
The parametric equations are then:
- x = 1 + 2t
- y = 4 + t
- z = 2 - 2t
For the symmetric equations, we use the direction vector and the initial point to form the following equations:
\(\frac{x-1}{2} = \frac{y-4}{1} = \frac{z-2}{-2}\)
Note: When writing the symmetric equation, ensure that each fraction is equal to the parameter t.