Final answer:
To create the vector equation for the line segment from (2, -1, 8) to (5, 5, 5), you find the directional vector by subtracting the first point from the second and then express the initial point as a vector, resulting in the equation r(t) = (2i - j + 8k) + t(3i + 6j - 3k).
Step-by-step explanation:
To find a vector equation for the line segment from (2, −1, 8) to (5, 5, 5) with the parameter t, we first need to find the directional vector that points from the first point to the second. This directional vector is obtained by subtracting the coordinates of the first point from the coordinates of the second point. Therefore, the directional vector is (5 - 2, 5 - (-1), 5 - 8) = (3, 6, -3).
Since a line in three dimensions can be represented by a vector equation in the form r(t) = r0 + tδ, where r0 is the position vector of the initial point, you start by converting the first point (2, -1, 8) into its vector form, which is 2i - j + 8k. Then, we adjust the equation to include the parameter t and the directional vector. Thus, we have:
r(t) = (2i - j + 8k) + t(3i + 6j - 3k)
This is the vector equation for the line segment from (2, -1, 8) to (5, 5, 5) as a function of t, where t ranges from 0 to 1 to get all points on the segment.