Application of the cross product: max height of z?

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Homework Help Overview

The problem involves determining the maximum height (z) at which two conduits can pass without interference, given their diameters and spatial arrangement. The conduits are represented by lines AB and CD, with specific dimensions and a requirement that one conduit must pass under the other.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the cross product of vectors AB and CD to find a unit vector, and then dotting this with another vector to determine distances. There is mention of needing to find the shortest distance and how to set up equations involving the height z.

Discussion Status

Some participants have begun to articulate their thought processes and approaches, indicating progress in understanding the problem. Suggestions have been made regarding the use of perpendicular distances and the need to select appropriate solutions for z based on the context of the problem.

Contextual Notes

Participants note the specific diameters of the conduits and the requirement that one conduit must pass under the other, which may influence the calculations and assumptions made in the discussion.

brinethery
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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.
 
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brinethery said:

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".
 
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.
 

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