Final answer:
The distance between the skew lines r(t) = ⟨-1, -5, -4⟩t and p(s) = ⟨-1, -5, -2⟩s is 0 units. The two lines are either parallel or intersect at some point along their length.
Step-by-step explanation:
To find the distance between the skew lines r(t) = ⟨-1, -5, -4⟩t and p(s) = ⟨-1, -5, -2⟩s, we can use the distance formula. The distance between two skew lines can be found by finding the shortest distance between any two points on the lines.
Step 1: Choose a point on one line, let's say r(t), and a point on the other line, p(s).
Step 2: Find the vector connecting these two points, which is given by the difference between the two points.
Step 3: Find the magnitude of this vector to get the distance between the two lines.
In this case, we have r(t) = ⟨-1, -5, -4⟩t and p(s) = ⟨-1, -5, -2⟩s. Let's choose t = 0 and s = 0 for convenience.
Substituting t = 0 and s = 0 into the equations, we get r(0) = ⟨-1, -5, -4⟩(0) = ⟨0, 0, 0⟩ and p(0) = ⟨-1, -5, -2⟩(0) = ⟨0, 0, 0⟩.
The vector connecting these two points is ⟨0, 0, 0⟩ - ⟨0, 0, 0⟩ = ⟨0, 0, 0⟩.
The magnitude of this vector is |⟨0, 0, 0⟩| = 0.
Therefore, the distance between the skew lines r(t) = ⟨-1, -5, -4⟩t and p(s) = ⟨-1, -5, -2⟩s is 0 units. This means that the two lines are either parallel or intersect at some point along their length.