Final answer:
The point of intersection for the given lines is (-136/27, 13/3, 11/3).
Step-by-step explanation:
In order to determine whether the lines x = p + su and x = q + tv intersect, we need to find the values of s and t that make the equations true. We can do this by setting the equations equal to each other:
p + su = q + tv
Now, we can isolate the variables:
- s = (q - p + tv) / u
- t = (p - q + su) / v
Substituting the given values, we have:
- s = (8 - (-1) + (−1)v) / 1
- t = ((−1) - 8 + (1)(8u)) / (-1)
Simplifying each expression, we get:
- s = (9 - v) / 1
- t = (−9 + 8u) / (-1)
Since we want the lines to intersect, there must be values of s and t that satisfy both equations. By substituting the expressions for s and t in terms of v and u into the equation for x, we can find the point of intersection:
- x = p + su = -1 + (9 - v) / 1 u = 8 - v
- x = q + tv = 8 + (−9 + 8u) / (-1) v = 8u - 17
By setting the expressions for x equal to each other, we have:
8 - v = 8u - 17
Now, we can solve for v in terms of u:
v = 8u - 25
Since u = 8 - v, we can substitute that expression into the equation:
v = 8(8 - v) - 25
Expanding and simplifying, we get:
v = 64 - 8v - 25
Combining like terms, we have:
9v = 39
Dividing both sides by 9, we find:
v = 13/3
Substituting this value back into the equation v = 8u - 25, we can solve for u:
13/3 = 8u - 25
Adding 25 to both sides, we have:
88/3 = 8u
Dividing both sides by 8, we get:
11/3 = u
Now that we have the values of u and v, we can substitute them back into either of the expressions for x to find the point of intersection. Substituting into x = p + su, we get:
x = -1 + (9 - 13/3)(11/3) = -1 + (9 - 143/9)/3 = -1 + 72/9 - 143/27 = -1 + 8 - 143/27 = -136/27
Therefore, the point of intersection is (-136/27, 13/3, 11/3).