GCSE-readyVectors: intersecting lines and equating coefficients
To find where two lines in a vector diagram meet, write the same vector two ways and compare the amounts of each base vector.
The key fact
This works because you cannot make a non-zero multiple of a out of b.
3a + (k − 1)b = (m + 1)a + 5b gives 3 = m + 1 and k − 1 = 5.
Two routes to one point
On BM: OX = b + μ(12a − b) = μ2a + (1 − μ)b
On AN: OX = a + λ(23b − a) = (1 − λ)a + 2λ3b
Equate: μ2 = 1 − λ and 1 − μ = 2λ3.
Solve and interpret
So OX = 14a + 12b.
λ = 34 means AX = 34AN, so X divides AN in the ratio 3 : 1.
3a + (k − 1)b = (m + 1)a + 5b, where a and b are not parallel. Find k and m.
- Compare the a parts: 3 = m + 1, so m = 2.
- Compare the b parts: k − 1 = 5, so k = 6.
Answer: k = 6, m = 2
Write the vector to the meeting point along both lines with unknowns λ and μ, then equate the coefficients of a and of b, because a and b are not parallel.
Try two quick questions
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2 more questions
Try each one, then tap to check.
5a + (k + 2)b = (m − 1)a + 7b, where a and b are not parallel. Find k.
Show the answer
Answer: 5
k + 2 = 7, so k = 5.
5a + (k + 2)b = (m − 1)a + 7b, where a and b are not parallel. Find m.
Show the answer
Answer: 6
m − 1 = 5, so m = 6.
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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.