Let l1 be the line in xy-plane with x and y intercepts 18 and 142√ respectively, and l2 be the line in zx-plane with x and z intercepts −18 and −163√ respectively. If d is the shortest distance between the

To find the shortest distance between the lines l1 and l2, we can use the formula for the distance between two parallel lines in three-dimensional space.

First, let's find the direction vectors for both lines. The direction vector for l1 can be found by subtracting the coordinates of the x and y intercepts:

l1 direction vector = (0 - 18, 0 - 142√, 0) = (-18, -142√, 0)

Similarly, the direction vector for l2 can be found by subtracting the coordinates of the x and z intercepts:

l2 direction vector = (0 - (-18), 0, 0 - (-163√)) = (18, 0, -163√)

Next, we can find a vector that is perpendicular to both direction vectors. This can be done by taking the cross product of the two direction vectors:

perpendicular vector = l1 direction vector x l2 direction vector

Using the cross product formula, we have:

perpendicular vector = (-18, -142√, 0) x (18, 0, -163√)

To calculate the cross product, we can use the determinant method:

perpendicular vector = ((-142√)(-163√) - 0) i - ((0)(-163√) - (0)(18)) j + ((-18)(0) - (-142√)(18)) k = (23186) i + 0 j + (2556√) k = (23186, 0, 2556√)

Now, we can find the shortest distance between the lines by projecting the vector between any point on l1 and l2 onto the perpendicular vector. Let's choose the point (18, 0, 0) on l2.

The vector between the point (18, 0, 0) on l2 and any point (x, y, 0) on l1 is given by:

vector = (x - 18, y - 0, 0 - 0) = (x - 18, y, 0)

To find the projection of this vector onto the perpendicular vector, we can use the dot product:

projection = (x - 18, y, 0) · (23186, 0, 2556√)

Using the dot product formula, we have:

projection = (x - 18)(23186) + (y)(0) + (0)(2556√) = 23186(x - 18)

The shortest distance between the lines is the magnitude of the projection vector, divided by the magnitude of the perpendicular vector:

d = |projection| / |perpendicular vector| = |23186(x - 18)| / |23186, 0, 2556√|

Since the magnitude of the perpendicular vector is a constant, we can simplify the equation:

d = |x - 18| / √(23186^2 + 2556^2)

Therefore, the shortest distance between the lines l1 and l2 is |x - 18| / √(23186^2 + 2556^2).