Key Idea: A straight line in vector form is r = a + λd: a starting point plus a direction. Changing λ slides you along the line, for example along a boat's path.
Using the equation
- a is a point on the line; d is the direction, the part multiplied by λ.
- Through two points A and B: d = B − A, then r = A + λd.
- Put a number in for λ to find a position.
Is a point on the line?
- Set equal — Set r equal to the point.
- Solve — Solve one component for λ.
- Check — Put that λ into every other component.
- Decide — One λ fits all: on the line. Otherwise: off the line.
A point is on the line only if the same λ works everywhere.
Path r = (1, 4) + λ(2, −3). Is (7, −5) on it? Yes. λ = 3 fits both components.
Is (5, 0) on that path? 1 + 2λ = 5 gives λ = 2, but 4 − 3(2) = −2, not 0. So no.
A line goes through A(0, 1) and B(2, 5). Give a direction vector. B − A = (2, 4).
Where is the boat when λ = 1? (3, 1).