VEC016 · Vectors · Class 12 + B.Tech
3D vector line
A line in 3D is a point a plus any multiple t of a direction vector d.
r = a + td3D vector line
📐 Formula
r = a + t d
✏️ Worked example
r = (1, 2, 1) + t(2, −1, 1); t = 1 gives (3, 1, 2).
🌍 Where it is used
Laser paths, flight routes, robot arms.
❓ Quick check
For r = (0, 0, 0) + t(1, 2, 3), which point is on the line?
🏋️ Practice
EasyPoint at t = 0 on r = (1, 2, 1) + t(2, −1, 1)?
Answer: (1, 2, 1)
MediumDirection vector of the line through (1, 0, 0) and (3, 1, 2)?
Answer: (2, 1, 2)
ChallengePoint at t = −1 on the line above?
Answer: (−1, 3, 0)