Final answer:
To find the parametrization of a line given two points, subtract the starting point from the ending point to get the direction vector, and then use the starting point and the direction vector with a parameter t to express the line in vector form.
Step-by-step explanation:
To find a vector parametrization r(t) for the line that passes through the points (1, 1, 1) and (9, −7, 8), we first need to determine the direction vector of the line. We can do this by subtracting the coordinates of the starting point from the ending point:
(9 − 1, −7 − 1, 8 − 1) = (8, −8, 7).
Then, we can use one of the points and the direction vector to write the parametric equations of the line. Using the point (1, 1, 1) as the initial point, we have:
r(t) = (1, 1, 1) + t(8, −8, 7),
or, in component form:
r(t) = (1 + 8t)î + (1 − 8t)ç + (1 + 7t)k.
This gives us the vector parametrization of the line where t is the parameter.