Assignment on Lines

1. Find the vector and parametric equations for the line through the point P = (2, 5, −1) and parallel to the vector v = ⟨−3, 1, 2⟩.

2. Find the vector and parametric equations for the line through the point P = (5, 8, −6) and parallel to the vector v = 2i − 3j + 4k.

3. Find the vector and parametric equations for the line through the points P = (4, 1, −8) and Q = (2, 3, 5).

4. Find the angle between the lines l_{1} and l_{2} given by:

l_{1} : r = ⟨1 − 2t, 3 + t, 4 − 5t⟩ and l_{2} : r = ⟨2 − s, 1 − 2s, 3 + 2s⟩.

5. Find the parametric equations of the line through (3, −1, 2) and parallel to the line r = ⟨2 − 3t, 7 + t, 8 + 5t⟩.

6. Find the vector form of the line through the point (5, 2, −3) and orthogonal to the lines r = ⟨2 + t, 3 − 2t, 4 − 5t⟩ and r = ⟨1 − t, 2t, 3 + 4t⟩.

7. Determine whether the lines l_{1} and l_{2} are parallel, skew, or intersecting. If they intersect, find their point of intersection.

(a) l_{1}: x = 4 − t, y = 2t, z = 3 + 4t, and l_{2}: x = 2 + 3s, y = 1 − s, z = 4 + s.

(b) l_{2}: r = ⟨3 − 4t, 2 + t, 2t⟩, and r = ⟨3 + 2s, 1 − s, 8 + 3s⟩