Find a point of intersection of the lines {p:p = (1, -5, 2) + t(-1, 1, 0)} and {p:p = (3, -3, 1) + t(4, 0, -1)}
To find the point of intersection of the two lines, we need to find values of t for which the position vectors of the two lines are equal.
Let's set the position vectors equal to each other and solve for t:
(1, -5, 2) + t(-1, 1, 0) = (3, -3, 1) + t(4, 0, -1)
Let's set up equations for each component:
1 - t = 3 + 4t
-5 + t = -3
2 = 1 - t
From the second equation, we have t = -2. Substituting this value into the other equations, we get:
1 - (-2) = 3 + 4(-2)
-5 - 2 = -3
2 = 1 - (-2)
Simplifying, we have:
3 = -5
-7 = -3
2 = 3
These equations are inconsistent and have no solutions. Therefore, the two lines do not intersect and are parallel or skew.
To find the point of intersection of the given lines, we need to find the values of t for which the two lines coincide.
Let's equate the x, y, and z coordinates of the two lines and solve for t:
x-coordinate: 1 - t = 3 + 4t
=> t = -0.5
y-coordinate: -5 + t = -3
=> t = 2
z-coordinate: 2 + 0t = 1 - t
=> t = 0.5
Since we have different values of t, we can conclude that the lines do not intersect.