The shortest cable to an inclined roof diagonal

Q23 inclined rectangular roof and ground with dimensionsEFGHABCD10 m10 m5 m3 m3 m6 m6 m4 mOijk
Question diagram 2
Q23 inclined rectangular roof and ground with dimensionsEFGHABCD10 m10 m5 m3 m3 m6 m6 m4 mOijk

Native roof and ground geometry

The diagram above shows a partial design of the roof of the gallery of an amphitheatre. ABCDABCD is an inclined rectangular roof, where AB=10AB=10 m and BC=5BC=5 m. EFGHEFGH is a rectangle on a horizontal ground, where EF=10EF=10 m and EH=4EH=4 m. The points A,BA,B are 3 m directly above E,FE,F respectively. The points C,DC,D are 6 m directly above G,HG,H respectively.

Point OO is the centre of EFGHEFGH and is taken as the origin. Perpendicular unit vectors i,j,k\mathbf i,\mathbf j,\mathbf k are in the direction of EF→,EH→,EA→\overrightarrow{EF},\overrightarrow{EH},\overrightarrow{EA} respectively.

(i) Find a vector equation of the line passing through A,CA,C. marks

(ii) A spot light is fixed at the top of a vertical pillar erected at point OO with coordinates (15,6,0)(15,6,0). Find the height of the pillar if the top of the pillar is collinear with A,CA,C. marks

(iii) The main electrical supply to the gallery runs along the diagonal ACAC. Find the position vector of point XX on ACAC such that the length of the electric cable that is required to provide electricity from the main electrical supply to an amplifier mounted at DD, will be the least. marks

[2010/PJC/Prelim/II/3]

Total marks: 8

Solution

(i)

Line through two points

Find a vector equation of the line through A(−5,−2,3)A(-5,-2,3) and C(5,2,6)C(5,2,6).

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Hint

  1. AC→=(10,4,3)\overrightarrow{AC}=(10,4,3).

Answer · (i)

r=(−5,−2,3)+t(10,4,3)r=(-5,-2,3)+t(10,4,3).

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The roof and its diagonal AC

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

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(ii)

Collinearity with a pillar

A spotlight at the top of a vertical pillar at (15,6,0)(15,6,0) is collinear with AA and CC. Find the pillar's height.

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Hint

  1. The pillar top is (15,6,h)(15,6,h) and lies on line ACAC.

Answer · (ii)

9 m9\,\mathrm m.

The printed base name O conflicts with the origin O; interpret the base as the separately located point (15,6,0). Its top lies on the extended diagonal beyond C.

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The pillar meets the extended diagonal

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

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(iii)

Shortest distance to a line

Find XX on ACAC so that the cable from XX to the amplifier at DD is shortest.

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Hint

  1. X=(−5,−2,3)+λ(10,4,3)X=(-5,-2,3)+\lambda(10,4,3).

  2. DX→⋅AC→=0\overrightarrow{DX}\cdot\overrightarrow{AC}=0.

Answer · (iii)

X=(−3,−6/5,18/5)X=(-3,-6/5,18/5).

Its parameter 1/5 lies within the roof diagonal segment.

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Slide X along AC and measure the cable to D

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.