Final answer:
The parametric equations for the line through points P(-1, -1, 7) and Q(6, -6, -6) are x(t) = -1 + 7t, y(t) = -1 - 5t, and z(t) = 7 - 13t, where t is a parameter representing any real number.
Step-by-step explanation:
To find parametric equations for the line through points P(-1, -1, 7) and Q(6, -6, -6), we first need to find the direction vector by subtracting the coordinates of P from those of Q. This gives us the vector v = Q - P = (6 - (-1), -6 - (-1), -6 - 7) = (7, -5, -13).
Next, we use point P as a reference and express the parametric equations as follows:
- x = x1 + tvx = -1 + 7t
- y = y1 + tvy = -1 - 5t
- z = z1 + tvz = 7 - 13t
Here, t is the parameter, which can take any real number value. The parametric equations of the line are therefore x(t) = -1 + 7t, y(t) = -1 - 5t, and z(t) = 7 - 13t.