Example 1-6-5 - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Education : Study Guides : MultivariateCalculus : Chapter 1 : Examples : Section 1-6 : Example 1-6-5

Chapter 1: Vectors, Lines and Planes

Section 1.6: Lines

Example 1.6.5

Lines $L_{1}$ and $L_{2}$ both have the common direction $V = 3 i - 2 j + 5 k$ , with $L_{1}$ passing through point P: $(7, - 4, 6)$ and $L_{2}$ passing through Q: $(4, - 7, 1)$ .

a)	Find the equations of $L_{1}$ and $L_{2}$ .

b)	Calculate the distance between $L_{1}$ and $L_{2}$ .

Solution

Mathematical Solution

Part (a)

If P and Q are position vectors to points P and Q, respectively, then

line $L_{1}$ is described vectorially by $R = P + t V$ , that is, by $[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 7 \\ - 4 \\ 6 \end{array}] + t [\begin{array}{c} 3 \\ - 2 \\ 5 \end{array}]$ and

line $L_{2}$ is described vectorially by $R = Q + s V$ , that is, by $[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 4 \\ - 7 \\ 1 \end{array}] + s [\begin{array}{c} 3 \\ - 2 \\ 5 \end{array}]$ .

Part (b)

•	Figure 1.6.5(a) is a sketch of the parallel lines $L_{1}$ and $L_{2}$ , along with the points P and Q, the direction vector V, the vector U from point Q to point P, and $U_{⊥ V}$ , the component of U orthogonal to V.

•	The distance between the points is the magnitude of $U_{⊥ V} = U - U_{V}$ , where $U_{V}$ is the projection of U onto V.

•	Obtain the vector

$U = P - Q = [\begin{array}{c} 7 \\ - 4 \\ 6 \end{array}] - [\begin{array}{c} 4 \\ - 7 \\ 1 \end{array}] = [\begin{array}{c} 3 \\ 3 \\ 5 \end{array}]$

Figure 1.6.5(a) Parallel Lines $L_{1}$ and $L_{2}$ , and vectors $U = P - Q$ and V

then calculate

$U_{V} = \frac{U \cdot V}{V \cdot V} V = \frac{28}{38} V = \frac{14}{19} [\begin{array}{c} 3 \\ - 2 \\ 5 \end{array}]$ and $U_{⊥ V} = U - U_{V} = [\begin{array}{c} 3 \\ 3 \\ 5 \end{array}] - \frac{14}{19} [\begin{array}{c} 3 \\ - 2 \\ 5 \end{array}] = \frac{5}{19} [\begin{array}{c} 3 \\ 17 \\ 5 \end{array}]$

so that finally, the distance between the lines is

$∥U_{⊥ V}∥$ = $\frac{5}{19} \sqrt{9 + 289 + 25} = 5 \sqrt{323} / 19$

Maple Solution - Interactive

Part (a)

•	Tools≻Load Package: Student Multivariate Calculus

Loading Student:-MultivariateCalculus

•	Enter V as per Table 1.1.1.

•	Context Panel: Assign to a Name≻V

$⟨3, - 2, 5⟩$ $\overset{assign to a name}{\to}$ $V$

Obtain lines $L_{1}$ and $L_{2}$

•	Form a sequence of point P (or Q) and the name V.

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line

•	Context Panel: Assign to a Name≻L[k], $k = 1, 2$

$[7, - 4, 6], V$ $\overset{make line}{\to}$ $<< Line 1 >>$ $\overset{assign to a name}{\to}$ $L_{1}$

$[4, - 7, 1], V$ $\overset{make line}{\to}$ $<< Line 2 >>$ $\overset{assign to a name}{\to}$ $L_{2}$

Display a representation of each line

•	Write the name of the line. Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻vectors (Use $t$ on $L_{1}$ and $s$ on $L_{2}$ as the parameters along the lines.)

$L_{1}$ = $<< Line 1 >>$ $\overset{representation}{\to}$

$L_{2}$ = $<< Line 2 >>$ $\overset{representation}{\to}$

Part (b)

•	Write the sequence of names for the two lines. Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance

•	Context Panel: Combine≻radical

•	Context Panel: Approximate≻5 (digits)

$L_{1}, L_{2}$ = $<< Line 1 >>, << Line 2 >>$ $\overset{distance}{\to}$ $\frac{5}{38} \sqrt{34} \sqrt{38}$ $\overset{combine}{=}$ $\frac{5}{19} \sqrt{323}$ $\overset{at 5 digits}{\to}$ $4.7295$

The exact distance between the lines can also be expressed as $5 \sqrt{17 / 19}$ , or even $5 \sqrt{323} / 19$ .

The traditional approach to the calculation of the distance between two parallel lines is vector-based, obtaining, for any vector between the two lines, its scalar projection orthogonal to the common direction of the lines. (See Figure 1.6.5(a).) This calculation is give below.

Define the position vectors P and Q

•	Enter P as per Table 1.1.1.

•	Context Panel: Assign to a Name≻P

$⟨7, - 4, 6⟩$ $\overset{assign to a name}{\to}$ $P$

•	Enter Q as per Table 1.1.1.

•	Context Panel: Assign to a Name≻Q

$⟨4, - 7, 1⟩$ $\overset{assign to a name}{\to}$ $Q$

Obtain U, the vector from Q on $L_{2}$ to P on $L_{1}$

•	Write the definition of U.

•	Context Panel: Assign Name

$U = P - Q$ $\overset{assign}{\to}$

Obtain $U_{V}$ , the projection of U upon V

•	Write the sequence of names U, V. Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection

•	Context Panel: Assign to a Name≻UV

$U, V$ = $\overset{projection}{\to}$ $\overset{assign to a name}{\to}$ $UV$

Obtain $U_{⊥ V} = U - U_{V}$ and its magnitude

•	Write the difference of U and $U_{V}$ . Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Norm

$U - UV$ = $\overset{norm}{\to}$ $\frac{5}{19} \sqrt{323}$

Maple Solution - Coded

Part (a)

If P and Q are position vectors to points P and Q, respectively, then

Part (b)

Initialize

•	Install the Student MultivariateCalculus package.

$with (Student :- MultivariateCalculus) &colon;$

•	Define the position vectors P and Q, and the direction vector V.

$P, Q, V ≔ ⟨7, - 4, 6⟩, ⟨4, - 7, 1⟩, ⟨3, - 2, 5⟩ &colon;$

•	Obtain $Q$ , the vector from Q to P.

$U ≔ P - Q &colon;$

Calculate $∥U_{⊥ V}∥$ = $∥U - U_{V}∥$

•	Apply the Norm and DotProduct commands.

$Norm (U - \frac{DotProduct (U, V)}{DotProduct (V, V)} V)$ = $\frac{5}{19} \sqrt{323}$

<< Previous Example Section 1.6 Next Section >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com

Download Help Document

Maple

Maple Add-Ons

Student Success Platform

Improving Retention Rates

MapleSim

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim

Maple

Powerful math software that is easy to use

Maple Add-Ons

Student Success Platform

Improving Retention Rates

MapleSim

Advanced System Level Modeling

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim