To find the distance between the two skew lines, we need to find a vector connecting the two lines, and then find the projection of the vector connecting any point on one line to the other line.
Let’s first find a vector connecting the two lines. We can take any two points, one on each line, and subtract their position vectors. Let’s take the point (1, 2, -4) on the first line and the point (3, 3, -5) on the second line:
v = (3i + 3j – 5k) – (i + 2j – 4k) = 2i + j – k
Next, we need to find the projection of this vector onto a direction vector of one of the lines. Let’s take the direction vector of the first line, which is (2i + 3j + 6k). We can use the projection formula:
proj_w v = ((v . w) / ||w||^2) w
where w is the direction vector of the first line.
First, we need to find the dot product v . w:
v . w = (2i + j – k) . (2i + 3j + 6k) = 4 + 3 – 6 = 1
Next, we need to find the magnitude of w:
||w|| = sqrt((2^2) + (3^2) + (6^2)) = sqrt(49) = 7
Finally, we can plug in these values into the projection formula:
proj_w v = ((v . w) / ||w||^2) w = (1 / (7^2)) (2i + 3j + 6k) = (2/49)i + (3/49)j + (6/49)k
This projection vector represents the distance between the two lines. The magnitude of this vector is the distance between the two lines:
distance = ||proj_w v|| = sqrt((2/49)^2 + (3/49)^2 + (6/49)^2) = sqrt(49)/49 = 1/7
Therefore, the distance between the two lines is 1/7.
