Concept explainers
Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes?
Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming
product.
The symmetric forms of two lines,
You are to develop a formula for the distance
minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines.
9. Consider the following application. Engineers at a refinery have determined they need to install support struts between many of the gas pipes to reduce damaging vibrations. To minimize cost, they plan to install these struts at the closest points between adjacent skewed pipes. Because they have detailed schematics of the structure, they are able to determine the correct lengths of the struts needed, and hence manufacture and distribute them to the installation crews without spending valuable time making measurements. The rectangular frame structure has the dimensions
Figure 2.74 Two pipes cross through a standard frame unit. Write down the vectors along the lines representing those pipes, find the cross product between them from which to create the unit vector
Want to see the full answer?
Check out a sample textbook solutionChapter 2 Solutions
Calculus Volume 3
Additional Math Textbook Solutions
Mathematics for Elementary Teachers with Activities (5th Edition)
Thinking Mathematically (7th Edition)
Thinking Mathematically (6th Edition)
Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
- Laying Phone Cable City A lies on the north bank of a river that is 1 mile wide. You need to run a phone cable from City A to City B, which lies on the opposite bank 5 miles down the river. You will lay L miles of the cable along the north shore of the river, and from the end of that stretch of cable you will lay W miles of cable running under water directly toward City B. See Figure 2.108. Figure 2.108 You will need the following fact about right triangles: A right triangle has two legs, which meet at the right angle, and the hypotenuse, which is the longest side. An ancient and beautiful formula, the Pythagorean theorem, relates the lengths of the three sides: Lengthofhypotenuse=Lengthofoneleg2+Lengthofotherleg2 a. Find an appropriate right triangle that shows that W=1+(5L)2 b. Find a formula for the length of the total phone cable P from City A to City B as a function of L. c. Make a graph of the total phone cable length P as a function of L, and explain what the graph is showing. d. What value of L gives the least length for the total phone cable? Draw a picture showing the least-length total phone cable.arrow_forwardA parabolic reflector for a car headlight forms a bowl shape that is 6 in. wide at its opening and 3 in. deep, as shown in the figure at the left. How far from the vertex should the filament of the bulb be placed if it is to be located at the focus?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781305652224Author:Charles P. McKeague, Mark D. TurnerPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning