Multiple choice.

How would I answer this in a short way?
what is the distance between the skew lines L1:(0,4,-3)+s(-1,1,3),seR and L2: (1,2,5)+t(-3,2,5),teR?

a) 4 root 2
b) 7/2
c) root 2
d) (7 root 2)/2

as your text no doubt shows, given the two lines

L1 = a+bs
L2 = c+dt

the distance between the lines is given by

(a-c).(bxd)/|bxd|

In this case, that is

(-1,2,-8).(-1,1,3)x(-3,2,5)/|(-1,1,3)x(-3,2,5)| = 15/3√2 = 5/√2

Hmmm. Better check my math.

To Sara:

I got 5/√2 as well. Perhaps it is time to check if there are typos in the question or the answer.

Also, to print s∈R, you can write
s & i s i n ; R, but skip the spaces.

To find the distance between two skew lines, you can use the formula:

d = |(L1 - L2) · n| / |n|

where L1 and L2 are any points on the two lines, · denotes the dot product, and n is the direction vector of one of the lines.

For the given lines L1: (0,4,-3) + s(-1,1,3) and L2: (1,2,5) + t(-3,2,5), we can choose points (0,4,-3) and (1,2,5) on L1 and L2 respectively.

First, let's find the direction vector n of L1. n can be found by taking the coefficients of s in L1: (-1,1,3).

Next, calculate the vector difference L1 - L2, which is obtained by subtracting the respective coordinates of the chosen points.

Now, take the dot product of (L1 - L2) and n, and divide it by the magnitude of n, to find the distance between the skew lines.

Finally, compare the calculated distance with the choices given, and choose the option that matches.

Remember to simplify any square roots in your final answer.