Question

Determine whether the lines intersect. If they do, find the coordinates of the point of intersection.
[x, y, z]=[2, 3, 2] + s[3, 4, 10]
[x, y, z]=[3, 4, 2] + t[4, 5, 3]

Answer 1

For line 1 : x = 2+3s, y = 3+4s, z = 2+10s
For line 2 : x = 3+4t, y = 4+5t, z = 2+3t

Let the two lines intersect. This means that when we solve for s and t, we will get a unique result.
2+3s = 3+4t .... (1)
3+4s = 4+5t .... (2)
2+10s = 2+3t .... (3)

From (3),
10s = 3t or s = 3t/10
substituting in (1), we get
9t/10 = 1 + 4t
=> 31t/10 = -1
=> t = -10/31
s = 3t/10 = -3/31
So from (1) and (3), we get
s = -3/31, t = -10/31

This value of s and t should also hold true for (2) if there is a point of intersection
substituting the values in (2), we get
LHS = 3 + 4(-3/31), RHS = 4 + 5(-10/31)
LHS = 81/31, RHS = 74/31
As LHS is not equal to RHS,
there is no point of intersection