Answer:
d = 2 / √19
Explanation:
To determine the distance between two lines, we can find the shortest distance between any two points on the lines. In this case, we'll find the shortest distance between the lines T = (0,1,-1) + s(3,0,1) and T = (0,0,1) + t(1,1,0).
Let's denote the first line as L1 and the second line as L2.
The direction vectors of the lines are:
L1: (3, 0, 1)
L2: (1, 1, 0)
To find the shortest distance, we can take a point on each line and calculate the vector connecting them. We'll choose the points P1(0, 1, -1) on L1 and P2(0, 0, 1) on L2.
The vector connecting P1 and P2 is given by P2 - P1:
V = (0, 0, 1) - (0, 1, -1)
= (0, -1, 2)
To find the distance between the lines, we need to find the component of vector V that is perpendicular to both L1 and L2. This can be done using the dot product.
The dot product of V with the direction vectors of the lines is:
V · L1 = (0, -1, 2) · (3, 0, 1) = 0 + 0 + 2 = 2
V · L2 = (0, -1, 2) · (1, 1, 0) = 0 - 1 + 0 = -1
The shortest distance between the lines is given by the absolute value of the ratio of the dot product results divided by the magnitude of the cross product of the direction vectors.
Magnitude of the cross product of L1 and L2:
|L1 x L2| = |(3, 0, 1) x (1, 1, 0)|
The cross product of L1 and L2 is given by:
L1 x L2 = (0 - 1, -3 - 0, 3 - 0) = (-1, -3, 3)
The magnitude of the cross product is:
|L1 x L2| = sqrt((-1)^2 + (-3)^2 + 3^2) = sqrt(1 + 9 + 9) = sqrt(19)
Therefore, the shortest distance between the lines L1 and L2 is:
d = |(V · L1) / |L1 x L2|| = |2 / sqrt(19)| = 2 / sqrt(19)
So, the distance between the lines is 2 / sqrt(19) units.