Final answer:
To determine whether the lines L₁ and L₂ intersect, are skew, or are parallel, we compare their direction vectors. By comparing the direction vectors, we find that the lines are not parallel. We can then solve the equations for x, y, and z to find the point of intersection.
Step-by-step explanation:
To determine whether the lines L₁ and L₂ intersect, are skew, or are parallel, we need to compare their direction vectors. To do this, we can equate the direction vector of L₁ (4,4,3) to the direction vector of L₂ (5,6,6) and check if they are proportional. If they are proportional, the lines are parallel. If they are not proportional, we can calculate the point of intersection by setting the x, y, and z components of L₁ and L₂ equal to each other and solving for t.
By comparing the direction vectors, we can see that the lines are not parallel. To find the point of intersection, we can set the x, y, and z components of L₁ and L₂ equal to each other:
x = 16 + 4t = -6 + 5t
y = 12 + 4t = -12 + 6t
z = 11 + 3t = -10 + 6t
Solving these equations simultaneously, we find that t = 1. Substituting this value of t into any of the equations, we can find the corresponding values of x, y, and z. In this case, x = 16 + 4(1) = 20, y = 12 + 4(1) = 16, and z = 11 + 3(1) = 14.