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Interpreting vector parameters as ratios

Interpreting vector parameters as ratios.

  • Write two routes to the same point and collect the a- and -components.
  • If a and are non-parallel, their coefficients must match separately.
  • Solve the resulting equations with a labelled operation at each step. A parameter is a fraction of its whole directed segment.

Example 1

The vectors a and are non-parallel. and . D lies on AB with AD:DB = 1:1. E lies on OB with OE = . OD and AE meet at P. Let and . Find OP:PD and AP:PE.

Two routes to the same intersectionOABDEPDiagram not to scale
D divides AB in the stated ratio.Route 1: O → P along OD.Find the whole vector from A to E.Route 2: O → A → P.Collect the a- and -components of route 2.Compare the a-components.Compare the -components.Make μ the subject of the second equation.Substitute into the first equation.Collect the λ terms on the left.Divide by the full coefficient of λ.Substitute λ back into μ = .λ is the fraction of the whole OD, so the remainder is 1 − λ.μ is the fraction of the whole AE.

Example 2

The vectors a and are non-parallel. and . D lies on AB with AD:DB = 2:1. E lies on OB with OE = . OD and AE meet at P. Let and . Find OP:PD and AP:PE.

Two routes to the same intersectionOABDEPDiagram not to scale
D divides AB in the stated ratio.Route 1: O → P along OD.Find the whole vector from A to E.Route 2: O → A → P.Collect the a- and -components of route 2.Compare the a-components.Compare the -components.Make μ the subject of the second equation.Substitute into the first equation.Collect the λ terms on the left.Divide by the full coefficient of λ.Substitute λ back into μ = .λ is the fraction of the whole OD, so the remainder is 1 − λ.μ is the fraction of the whole AE.

Your turn

Question 1

The vectors a and are non-parallel. and . D lies on AB with AD:DB = 3:1. E lies on OB with OE = . OD and AE meet at P. Let and . Find OP:PD and AP:PE.

Two routes to the same intersectionOABDEPDiagram not to scale
Check answer
D divides AB in the stated ratio.Route 1: O → P along OD.Find the whole vector from A to E.Route 2: O → A → P.Collect the a- and -components of route 2.Compare the a-components.Compare the -components.Make μ the subject of the second equation.Substitute into the first equation.Collect the λ terms on the left.Divide by the full coefficient of λ.Substitute λ back into μ = .λ is the fraction of the whole OD, so the remainder is 1 − λ.μ is the fraction of the whole AE.
Question 2

The vectors a and are non-parallel. and . D lies on AB with AD:DB = 4:1. E lies on OB with OE = . OD and AE meet at P. Let and . Find OP:PD and AP:PE.

Two routes to the same intersectionOABDEPDiagram not to scale
Check answer
D divides AB in the stated ratio.Route 1: O → P along OD.Find the whole vector from A to E.Route 2: O → A → P.Collect the a- and -components of route 2.Compare the a-components.Compare the -components.Make μ the subject of the second equation.Substitute into the first equation.Collect the λ terms on the left.Divide by the full coefficient of λ.Substitute λ back into μ = .λ is the fraction of the whole OD, so the remainder is 1 − λ.μ is the fraction of the whole AE.
Question 3

The vectors a and are non-parallel. and . D lies on AB with AD:DB = 1:2. E lies on OB with OE = . OD and AE meet at P. Let and . Find OP:PD and AP:PE.

Two routes to the same intersectionOABDEPDiagram not to scale
Check answer
D divides AB in the stated ratio.Route 1: O → P along OD.Find the whole vector from A to E.Route 2: O → A → P.Collect the a- and -components of route 2.Compare the a-components.Compare the -components.Make μ the subject of the second equation.Substitute into the first equation.Collect the λ terms on the left.Divide by the full coefficient of λ.Substitute λ back into μ = .λ is the fraction of the whole OD, so the remainder is 1 − λ.μ is the fraction of the whole AE.
Question 4

The vectors a and are non-parallel. and . D lies on AB with AD:DB = 2:2. E lies on OB with OE = . OD and AE meet at P. Let and . Find OP:PD and AP:PE.

Two routes to the same intersectionOABDEPDiagram not to scale
Check answer
D divides AB in the stated ratio.Route 1: O → P along OD.Find the whole vector from A to E.Route 2: O → A → P.Collect the a- and -components of route 2.Compare the a-components.Compare the -components.Make μ the subject of the second equation.Substitute into the first equation.Collect the λ terms on the left.Divide by the full coefficient of λ.Substitute λ back into μ = .λ is the fraction of the whole OD, so the remainder is 1 − λ.μ is the fraction of the whole AE.
Question 5

The vectors a and are non-parallel. and . D lies on AB with AD:DB = 3:2. E lies on OB with OE = . OD and AE meet at P. Let and . Find OP:PD and AP:PE.

Two routes to the same intersectionOABDEPDiagram not to scale
Check answer
D divides AB in the stated ratio.Route 1: O → P along OD.Find the whole vector from A to E.Route 2: O → A → P.Collect the a- and -components of route 2.Compare the a-components.Compare the -components.Make μ the subject of the second equation.Substitute into the first equation.Collect the λ terms on the left.Divide by the full coefficient of λ.Substitute λ back into μ = .λ is the fraction of the whole OD, so the remainder is 1 − λ.μ is the fraction of the whole AE.
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