Question

Find the distance D between the point (1, 4, 2) and the line passing through the points (−1, 3, −1) and (1, 2, 3).

Answer 1

it says the answer is sqrt(23/7) but im not sure how it got that.

Answer 2

Here is one way:
call pt (-1,3,-1) p0 and pt (1,2,3) p1 and pt (1,4,2) p2

subtract p0 from the other two points: u= p1-p0 and q= p2-p0
this will give you two vectors with a common origin
normalize the vector u to unit length: u1=u/ | u |

do a dot product between u1 and q: u1 dot q = x
now find y: y= q- x*u1
y is perpendicular to u, and its length is the distance between point p2 and the line

|y| is the answer.

Answer 3

The good news is that procedure got 1.8127 which matches sqrt(23/7)
The bad news is I did not do it by hand, I don't know the details...

Answer 4

Here are some of the numbers

p2 = 1 4 2
p0 = -1 3 -1
p1 = 1 2 3

q = 2 1 3
u = 2 -1 4
length of u sqrt(21)
u1= u/sqrt(21)

b= dot(p,u1)
b =3.2733
x= b*u1
x = 1.4286 -0.7143 2.8571

y= q- x
y = 0.5714 1.7143 0.1429
length(y)= 1.8127