Example 1-6-5 - Maple Help

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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.

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

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