To determine whether Line 1 (L1) and Line 2 (L2) intersect, we need to check if there is a common point that lies on both lines.
Step 1: Find the parametric equations for L1 and L2.
Parametric equations for a line are defined as follows:
L1: x = x1 + t⋅a, y = y1 + t⋅b, z = z1 + t⋅c
L2: x = x2 + s⋅d, y = y2 + s⋅e, z = z2 + s⋅f
Given:
Q1 = (4, -1, -2) and Q2 = (1, 0, -1) are two points on L1, and
P1 = (-6, 21, -8) is a point on L2 with direction vector →d = [-3, 9, -3]^T.
Step 2: Calculate the direction vectors for L1 and L2.
To find the direction vectors, we subtract the coordinates of the initial and terminal points for each line.
For L1, direction vector →v1 = Q2 - Q1
→v1 = (1 - 4, 0 - (-1), -1 - (-2))
→v1 = (-3, 1, 1)
For L2, direction vector →v2 = →d
→v2 = [-3, 9, -3]^T
Step 3: Set up equations with the parametric form.
Equations for L1 and L2 are:
L1: x = 4 - 3t, y = -1 + t, z = -2 + t
L2: x = -6 - 3s, y = 21 + 9s, z = -8 - 3s
Step 4: Solve for a point of intersection, if it exists.
To find the point of intersection, we need to equate the corresponding components of L1 and L2. We set:
4 - 3t = -6 - 3s
-1 + t = 21 + 9s
-2 + t = -8 - 3s
Solving these equations will give us the values of t and s. If t and s are both finite, then the lines intersect.
Using the first equation, we have: 3t - 3s = -10
Using the second and third equations, we have: t - 9s = 22 and t + 3s = -6
Solving this system of equations will give us the values of t and s.
Step 5: Substitute the values of t and s to find the point of intersection.
Once we find the values of t and s, we can substitute them back into the parametric form of either L1 or L2 to find the corresponding (x, y, z) values.
For example, substituting the values of t and s into the parametric equation of L1 (x = 4 - 3t, y = -1 + t, z = -2 + t) will give us the coordinates of the point of intersection (Q).
Note that if the lines do not intersect, then the system of equations will have no solution, and there will be no point of intersection.
By following these steps, you can determine whether Lines 1 and 2 intersect and find the point of intersection if it exists.