Application of the cross product: max height of z?

[brinethery]

Homework Statement

http://www.scribd.com/doc/82645310

In Figure 3-31, the lines AB and CD are the center lines of two conduits 1 ft. and 2 ft. in diameter respectively. Determine the maximum value of z so that the two may pass without interference. Conduit CD must pass under AB.

Homework Equations

ABxCD

The Attempt at a Solution

ABxCD, and then make this a unit vector. Then dot this unit vector AC? I know how to find the shortest possible distance, but I don't have a clue how to do this type of problem.

 

[gneill]

Homework Statement

http://www.scribd.com/doc/82645310

In Figure 3-31, the lines AB and CD are the center lines of two conduits 1 ft. and 2 ft. in diameter respectively. Determine the maximum value of z so that the two may pass without interference. Conduit CD must pass under AB.

Homework Equations

ABxCD

The Attempt at a Solution

ABxCD, and then make this a unit vector. Then dot this unit vector AC? I know how to find the shortest possible distance, but I don't have a clue how to do this type of problem.

Your suggested method looks good. You'll end up with an equation in unknown height Z for the perpendicular distance. It'll have two solutions for Z and you'll have to pick the appropriate one. You should be able to find a description and examples if you do a web search on "Perpendicular Distance between two Skew Lines".

 

[brinethery]

Okay I've sort of figured it out. Hopefully, I'm getting somewhere with this:

ABxCD, and then make this a unit vector. Then dot this unit vector with vector AC.

The question asks "what is the maximum height z can be...". The distance from the center to the radius of the two conduits when they're touching is going to be 1.5ft (since the diameters are 1ft and 2ft respectively). This means that I'll take the two vectors I dotted and set them equal to 1.5ft. Then I'll solve for z.

 

[brinethery]

Here is what I have so far:

http://www.scribd.com/doc/82663003

 

[asteriatic]

The cross product is a mathematical operation that is used to calculate the vector perpendicular to two given vectors. In this scenario, the cross product can be used to determine the maximum height of z so that the two conduits can pass without interference.

To find the maximum height, we can use the cross product of vectors AB and CD. This will give us a vector that is perpendicular to both AB and CD, which represents the possible height of z.

To calculate the cross product, we can use the formula AB x CD = |AB||CD|sin(θ)n, where θ is the angle between the two vectors and n is the unit vector in the direction perpendicular to both AB and CD.

Since the conduits are passing under each other, the angle between AB and CD is 90 degrees. Therefore, the cross product simplifies to |AB||CD|n.

To find the maximum value of z, we need to find the length of the cross product vector. This can be done by taking the magnitude of the vector, which is given by |AB||CD|.

Since the diameter of conduit AB is 1 ft and the diameter of conduit CD is 2 ft, the lengths of vectors AB and CD are 0.5 ft and 1 ft respectively.

Therefore, the maximum value of z can be calculated as z = |AB||CD| = (0.5 ft)(1 ft) = 0.5 ft.

This means that the maximum height of z should be 0.5 ft so that the two conduits can pass without interference. Any value of z greater than 0.5 ft would result in the conduits intersecting and causing interference.

In summary, the application of the cross product allows us to determine the maximum height of z so that the two conduits can pass without interference. This is a useful tool for engineers and scientists in designing structures and systems that need to avoid interference.