In r = p + λd, the vector p says where the line passes through and d says which way it runs. Any nonzero multiple of d gives the same direction, so two lines are parallel when their direction vectors are multiples of each other.
This lesson completes the toolkit for two-dimensional vector proofs.
How do you read a vector equation?
- Point: put λ = 0 and you get p, a point on the line.
- Direction: d is the direction. Simplify by dividing by a common factor.
- Gradient (for column vectors (x, y)): y ÷ x.
- Test a point: solve for λ from one coordinate, then confirm the other.
- Parallel test: check whether one direction is a multiple of the other.
Worked example
Line l: r = (2, −1) + λ(4, −6). (a) State a simpler direction vector and the gradient.
(b) Is (8, −10) on l? (c) Is (6, −8) on l?
(a) (4, −6) = 2(2, −3), so a simpler direction is (2, −3). The gradient is −3 ÷ 2 = −3/2.
(b) x: 2 + 4λ = 8, so λ = 3/2. y: −1 − 6λ = −1 − 9 = −10. It matches the y value, so (8, −10) is on l.
(c) x: 2 + 4λ = 6, so λ = 1. y: −1 − 6(1) = −7, but the point has y = −8. The two coordinates disagree, so (6, −8) is not on l.
The mistake to watch for
A common slip is to check only the x value.
Mistaken working: “For (6, −8): 2 + 4λ = 6, so λ = 1. A value of λ exists, so the point is on the line.”
The student never tested y.
A single coordinate can always be solved for some λ. The point is on the line only when the same λ fits both coordinates. Always carry λ across to the second coordinate.
Check yourself
1. Line r = (3, 1) + λ(−2, 5). State a direction vector with positive x part and the gradient.
Show answer
Multiply (−2, 5) by −1 to get (2, −5). Gradient = −5 ÷ 2.
Direction (2, −5), gradient −5/2
2. Are r = (1, 1) + λ(3, −2) and r = (0, 4) + μ(−6, 4) parallel? Are they the same line?
Show answer
(−6, 4) = −2(3, −2), so the lines are parallel. Is (0, 4) on the first? 1 + 3λ = 0 gives λ = −1/3, then y = 1 − 2(−1/3) = 5/3, not 4.
Parallel but not the same line.
3. The line r = (1, 0) + λ(2, k) passes through (7, 6). Find k.
Show answer
x: 1 + 2λ = 7, so λ = 3. y: 0 + 3k = 6.
k = 2
Where this leads next
Put all five skills together in the two-dimensional vector proofs practice set. You can revisit finding an intersection if the parameter work still needs time.
If you can read an equation but hesitate when planning a whole proof, our teachers can help through online one-to-one Additional Mathematics tuition.