Final answer:
The symmetric equations for the line are x - 1 = -1(t), y + 5 = 2(t), and z - 6 = -3(t).
Step-by-step explanation:
To find the symmetric equations for the line, we need to find the normal vector of the line first. Since the line is parallel to the vector <-1,2,-3>, the normal vector will be the same as the given vector. So, the normal vector is <-1,2,-3>.
Next, we substitute the coordinates of the given point (1,-5,6) into the equation of the line:
x - 1 = -1(t),
y + 5 = 2(t),
z - 6 = -3(t),
where t is a parameter representing the distance along the line.
These equations represent the symmetric equations for the line that passes through the point (1,-5,6) and is parallel to the vector <-1,2,-3>.