Find a point intersection of a lines {P:P = (1, -5, 2) + t(-1, 1, 0) & {P:p = (3, -3, 1) + t(4, 0, -1)}
To find the point of intersection, we need to find the values of t that satisfy both equations
(1, -5, 2) - t(1, -1, 0) = (3, -3, 1) + t(4, 0, -1)
Simplifying this equation, we get:
-2 - t = 4t
8t = -2
t = -1/4
We can substitute this value of t into either of the original equations to find the point of intersection. Using the first equation, we get:
P = (1, -5, 2) - (1/4)(-1, 1, 0)
P = (5/4, -21/4, 2)
Therefore, the point of intersection is P = (5/4, -21/4, 2).
NONSENCE!!!
the bot started with : (1, -5, 2) - t(1, -1, 0) = (3, -3, 1) + t(4, 0, -1)
you need a different parameter for the 2nd equation !!
so (1, -5, 2) - t(1, -1, 0) = (3, -3, 1) + k(4, 0, -1)
this gives us
1-t = 3+4k , -5+t = -3, 2 = 1-k
using the first two we get t=2, k=-1
this verifies the third equation
so the lines intersect at (-1 , -3, 2)
I apologize for my mistake. Thank you for correcting it. You are correct that a new parameter is needed for the second equation. Using the correct parameter, we get the intersection point (-1, -3, 2) as you have calculated.
To find the point of intersection between two lines, you need to set their equations equal to each other and solve for the values of t that make the equations true. Once you have the values of t, you can substitute them back into either equation to find the corresponding point.
Let's solve the given equations step by step:
Line 1: P = (1, -5, 2) + t(-1, 1, 0)
Line 2: P = (3, -3, 1) + t(4, 0, -1)
Setting these two equations equal to each other, we get:
(1, -5, 2) + t(-1, 1, 0) = (3, -3, 1) + t(4, 0, -1)
Now, let's equate the corresponding components:
x-coordinate: 1 - t = 3 + 4t
y-coordinate: -5 + t = -3
z-coordinate: 2 + 0t = 1 - t
Simplifying each equation:
-2t = 2 + 4t (subtract t from both sides of x-coordinate equation)
t = 2/6
t = 1/3
Substituting the value of t back into the y-coordinate equation:
-5 + (1/3) = -3
-5 + 1/3 = -3
-15/3 + 1/3 = -9/3
-14/3 = -9/3
Thus, we have found the value of t, which is 1/3.
Now, substituting t = 1/3 back into either of the original equations, let's find the corresponding point:
Using Line 1:
P = (1, -5, 2) + (1/3)(-1, 1, 0)
P = (1, -5, 2) + (-1/3, 1/3, 0)
P = (1 - 1/3, -5 + 1/3, 2 + 0)
P = (2/3, -14/3, 2)
Therefore, the point of intersection between the two lines is P = (2/3, -14/3, 2).