218k views
1 vote
Determine whether the lines x = p +su and x = q + tv intersect.p = −1, 8, 1 , q = 8, 8, 0 , u = 1, 8, −1 , v = −1, 1, 0,

The two lines intersect. The two lines do not intersect. Find the point of intersection. (If there is no solution, enter NO SOLUTION.)

[x, y, z] =

User Sergiom
by
7.7k points

1 Answer

2 votes

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:

  1. s = (q - p + tv) / u
  2. t = (p - q + su) / v

Substituting the given values, we have:

  1. s = (8 - (-1) + (−1)v) / 1
  2. t = ((−1) - 8 + (1)(8u)) / (-1)

Simplifying each expression, we get:

  1. s = (9 - v) / 1
  2. 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:

  1. x = p + su = -1 + (9 - v) / 1 u = 8 - v
  2. 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).

User Tonyjosi
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories