One λ must work for ALL coordinates: A point lies on the line only if a single value of λ reproduces every coordinate.
Is the point on the line?
Pick the easiest row
Solve it for λ.
Check the other rows
Put that same λ into every other row.
Decide
If they all match, the point is on the line; if even one disagrees, it isn't.
Going the other way: Going the other way is easier: to find the point at a particular λ, just substitute that number into r = a + λd.
IB-style question — does the point lie on the line?
A line has equation r = (1, 2, 0) + λ(2, −1, 3).
Determine whether the point Q(5, 0, 6) lies on the line.
Step by step
- Set r equal to Q and read off the three coordinate equations.
- Solve the first row for λ.
- Check λ = 2 in the other two rows.
Final answer
All three rows agree at λ = 2, so Q(5, 0, 6) lies on the line.
IB-style question — find the point at a given λ
For the line r = (4, −1, 2) + λ(1, 3, −2), find the position of the point when λ = 3.
Step by step
- Substitute λ = 3 into each coordinate.
- Multiply out and add.
Final answer
The point is (7, 8, −4).
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Meeting an axis = the other coordinates are 0: To find where a line crosses an axis, set the relevant coordinate equations to 0, solve for λ, then substitute back to get the crossing point.
| A point is on the… | when |
|---|---|
| x-axis | its y- and z-coordinates are both 0 |
| y-axis | x = z = 0 |
| z-axis | x = y = 0 |
In 2D: In 2D, a line meets the x-axis where y = 0 and the y-axis where x = 0.
IB-style question — where does the line cross an axis?
A 2D line has equation r = (6, 4) + λ(−2, 1).
Find where it crosses the x-axis.
Step by step
- On the x-axis the y-coordinate is 0. Set the y-row to 0.
- Substitute λ = −4 into the x-row to get the crossing point.
Final answer
The line crosses the x-axis at (14, 0).
Paper-2 model: a line that moves: On Paper 2 the same equation often describes a moving object (a boat, a drone, a helicopter):
r = (start position) + t(velocity), where the parameter t is time.
Then the velocity vector is d, and the speed is its magnitude |d|. Direction questions (e.g. an angle of descent) come from the components of d using trigonometry.
IB-style question — speed and angle of descent
A drone moves so that its position (in km) at time t hours is r = (1, 3, 2) + t(4, 0, −3).
(a) Find its speed. (b) Find the angle its path makes below the horizontal.
Step by step
- (a) Speed = magnitude of the velocity (4, 0, −3).
- (b) The descent is the downward part (z drops by 3) against the horizontal travel. Horizontal speed = √(4² + 0²) = 4.
- Take the inverse tangent.
Final answer
(a) Speed = 5 km/h. (b) Angle of descent ≈ 36.9° below the horizontal.