Coplanar lines and common point

Lines.

• L₁: (x − 1)/2 = (y + 1)/(−3) = (z + 10)/8, through A(1, −1, −10), direction (2, −3, 8).
• L₂: (x − 4)/1 = (y + 3)/(−4) = (z + 1)/7, through B(4, −3, −1), direction (1, −4, 7).

Coplanarity test

Two lines are coplanar if the determinant of AB, d₁, d₂ is zero. AB = (3, −2, 9).

| 3  −2   9 |
| 2  −3   8 |  = 3(−21 + 32) + 2(14 − 8) + 9(−8 + 3)
| 1  −4   7 |  = 3(11) + 2(6) + 9(−5)
                = 33 + 12 − 45 = 0

The determinant is zero, so the lines are coplanar.

Common point

General points: L₁: (1 + 2t, −1 − 3t, −10 + 8t); L₂: (4 + s, −3 − 4s, −1 + 7s). Equate:

1 + 2t = 4 + s          →  2t − s = 3
−1 − 3t = −3 − 4s       →  −3t + 4s = −2

From the first, s = 2t − 3. Put in the second: −3t + 8t − 12 = −2, so 5t = 10, t = 2 and s = 1.

Check the third coordinate: −10 + 8(2) = 6 and −1 + 7(1) = 6 ✓.

Point: x = 1 + 4 = 5, y = −1 − 6 = −7, z = 6.

Answer: the lines are coplanar and meet at (5, −7, 6).