A river as the intersection of two planes

Peter is using equations of planes to model two hillsides that meet along a river. The river is modelled by the line where the two planes meet.

Q21 hillsides H1 H2 meeting along river valleyH₁H₂
Question diagram 2
Q21 hillsides H1 H2 meeting along river valleyH₁H₂

Native source two hillsides meeting at river

One of the hillsides, H1H_1, contains the points A,B,CA,B,C with coordinates (3,0,2),(1,0,3),(2,−3,5)(3,0,2),(1,0,3),(2,-3,5) respectively. The point AA is on the river. The other hillside H2H_2 has equation 2x−y+kz=142x-y+kz=14, where kk is a constant.

(i) Find a vector equation of H1H_1 in scalar product form. marks

(ii) Show that k=4k=4 and deduce that point BB is also on the river. marks

(iii) Write down a cartesian equation of the river. mark

(iv) Show that BB is the point on the river that is nearest to CC. Hence find the exact shortest distance from CC to the river. marks

(v) Find the acute angle between BC,H2BC,H_2. marks

[2017/JJC/Prelim/II/3]

Total marks: 13

Solution

(i)

Plane through three points

H1H_1 contains A(3,0,2)A(3,0,2), B(1,0,3)B(1,0,3), C(2,−3,5)C(2,-3,5). Find its equation in scalar product form.

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Hint

  1. Normal =AB→×AC→=\overrightarrow{AB}\times\overrightarrow{AC}.

Answer · (i)

r⋅(3,5,6)=21r\cdot(3,5,6)=21.

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The hillside H₁ through A, B, C

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

(ii)

Common point

H2:2x−y+kz=14H_2:2x-y+kz=14 with AA on the river. Show k=4k=4 and that BB is on the river.

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Hint

  1. AA is on both planes.

  2. Test BB in both.

Answer · (ii)

k=4k=4; BB satisfies both plane equations.

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Both planes and the river

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

(iii)

Equation of the river

Write down a Cartesian equation of the river.

Hint

  1. Direction AB→=(−2,0,1)\overrightarrow{AB}=(-2,0,1).

Answer · (iii)

y=0y=0, (x−3)/(−2)=z−2(x-3)/(-2)=z-2.

(iv)

Nearest point

Show BB is the point on the river nearest to CC, and find the shortest distance.

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Hint

  1. BB is the nearest point iff BC→⋅AB→=0\overrightarrow{BC}\cdot\overrightarrow{AB}=0.

Answer · (iv)

AB⋅BC=0AB\cdot BC=0; nearest distance 14\sqrt{14}.

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Slide a point along the river

B and C stay fixed. Slide the trial point along the river; its distance to C is smallest when it reaches B.

Open the interactive view to explore this geometry in 3D.

(v)

Line meeting a plane

Find the acute angle between BCBC and H2H_2.

Hint

  1. Angle with the normal (2,−1,4)(2,-1,4), then the complement.

Answer · (v)

49.3∘49.3^\circ.