Vector-

**kingkaisai2** · July 14th 2006, 08:47 PM

A tunnel is to be evacated through a hill. In order to define position, coordinates (x,y,z) are taken relative to an origin O such that x is the distance east from O, y is the distance north and z is the verticl distance upwards, with one unit equal to 100m. The tunnel starts at point A(2,3,5) and runs in a straight line in the direction i+j-0.5k.

b) an old tunnel through the hill has equation r=4i+j+2k+x(7i+15j+0k). Show that the point p on the new tunnel where x=7.5 is directly above a point Q in the old tunnel. Find the separation PQ of the tunnels at this point.

**earboth** · July 17th 2006, 04:32 AM

Originally Posted by kingkaisai2

A tunnel is to be evacated through a hill. In order to define position, coordinates (x,y,z) are taken relative to an origin O such that x is the distance east from O, y is the distance north and z is the verticl distance upwards, with one unit equal to 100m. The tunnel starts at point A(2,3,5) and runs in a straight line in the direction i+j-0.5k.

b) an old tunnel through the hill has equation r=4i+j+2k+x(7i+15j+0k). Show that the point p on the new tunnel where x=7.5 is directly above a point Q in the old tunnel. Find the separation PQ of the tunnels at this point.

Hallo, kingkaisai2,

you want to calculate the shortest distance between 2 straight lines in $\mathbb{R}^3$ , which are not equal nor parallel nor intersecting. (The German word is "windschief", literally translated it means crooked. But I don't believe that this is the appropriate expression in English).

The equation of a line in 3D is: $g:\vec{x}=\vec{s}+r*\vec{u}$ , where the vector s determines the startingpoint S and the vector u the direction of the line.

The old tunnel is described by:
$o:\vec{x}=(2, 3, 5)+r*(1, 1, -0.5)$ and the new tunnel runs on this line:

$n:\vec{x}=(4, 1, 2)+r*(7, 15, 0)$

These lines are not parallel and they don't intersect. So the shortest distance is calculated by:

$d(o, n)=|\frac{(\overrightarrow{s_1}-\overrightarrow{s_2}) * (cross(\overrightarrow{u_1}, \overrightarrow{u_2}))}{|cross(\overrightarrow{u_1 },\overrightarrow{u_2})|}|$

Plug in the values you know:

$d(o, n)=\frac{((2, 3, 5)-(4, 1, 2))*(7.5, -3.5, 8)}{\sqrt{132.5}}\approx 0.17375$

Greetings

EB

**ThePerfectHacker** · July 17th 2006, 10:17 AM

Originally Posted by earboth

(The German word is "windschief", literally translated it means crooked. But I don't believe that this is the appropriate expression in English).

In America it is "skew".

**earboth** · July 17th 2006, 12:00 PM

Originally Posted by ThePerfectHacker

In America it is "skew".

Hallo, TPH,

thanks a lot. (I hope my description of the situation and my solution was understandable at least)

Greetings

EB

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