Question

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)}

Answer 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.