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Chapter 1: Vectors, Lines and Planes

Section 1.5: Applications of Vector Products

Example 1.5.11

Solve nine equations in nine unknowns to find the same set of reciprocal vectors that was found in Example 1.5.10.

Solution

Mathematical Solution

Respectively identifying $U_{k}, k = 1, 2, 3$ , with A, B, C, of Example 1.5.1, and taking the $V_{k}, k = 1, 2, 3$ , as the vectors

$[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]$

the equations $U_{i} \cdot V_{j} = δ_{ij}$ thereby generated are listed in Table 1.5.11(a). The solution of these equations is listed in Table 1.5.11(b). The resulting vectors $V_{k}$ are listed in Table 1.5.11(c).

$U_{1} \cdot V_{1} = δ_{11}$	$3 a_{1} - 2 a_{2} + 4 a_{3} = 1$
$U_{1} \cdot V_{2} = δ_{12}$	$3 b_{1} - 2 b_{2} + 4 b_{3} = 0$
$U_{1} \cdot V_{3} = δ_{13}$	$3 c_{1} - 2 c_{2} + 4 c_{3} = 0$
$U_{2} \cdot V_{1} = δ_{21}$	$2 a_{1} + 5 a_{2} - 4 a_{3} = 0$
$U_{2} \cdot V_{2} = δ_{22}$	$2 b_{1} + 5 b_{2} - 4 b_{3} = 1$
$U_{2} \cdot V_{3} = δ_{23}$	$2 c_{1} + 5 c_{2} - 4 c_{3} = 0$
$U_{3} \cdot V_{1} = δ_{31}$	$5 a_{1} + 7 a_{2} + 6 a_{3} = 0$
$U_{3} \cdot V_{2} = δ_{32}$	$5 b_{1} + 7 b_{2} + 6 b_{3} = 0$
$U_{3} \cdot V_{3} = δ_{33}$	$5 c_{1} + 7 c_{2} + 6 c_{3} = 1$
Table 1.5.11(a) The equations $U_{i} \cdot V_{j} = δ_{ij}$

$a_{1} = 29 / 97$

$a_{2} = - 16 / 97$

$a_{3} = - 11 / 194$

$b_{1} = 20 / 97$

$b_{2} = - 1 / 97$

$b_{3} = - 31 / 194$

$c_{1} = - 6 / 97$

$c_{2} = 10 / 97$

$c_{3} = 19 / 194$

Table 1.5.11(b) Solution of equations in Table 1.5.11(a)

$V_{1} = \frac{1}{194}$ $[\begin{array}{c} 58 \\ - 32 \\ - 11 \end{array}]$

$V_{2} = \frac{1}{194} [\begin{array}{c} 40 \\ - 2 \\ - 31 \end{array}]$

$V_{3} = \frac{1}{194} [\begin{array}{c} - 12 \\ 20 \\ 19 \end{array}]$

Table 1.5.11(c) Reciprocal vectors

Maple Solution - Interactive

Define the function $δ (i, j) = {\begin{array}{c} 1 & i = j \\ 0 & i \neq j \end{array}$

•	Expression palette: Piecewise template

•	Context Panel: Assign Function

$δ (i, j) = \{\begin{array}{c} 1 & i = j \\ 0 & i \neq j \end{array}$ $\overset{assign as function}{\to}$ $δ$

Define the vectors $U_{k}$ and $V_{k}$ , $k = 1, 2, 3$

•	Context Panel: Assign Name U is the list of vectors A, B, and C.

$U = [⟨3, - 2, 4⟩, ⟨2, 5, - 4⟩, ⟨5, 7, 6⟩]$ $\overset{assign}{\to}$

•	Context Panel: Assign Name V is the list of vectors $V_{1}$ , $V_{2}$ , $V_{3}$ .

$V = [⟨a_{1}, a_{2}, a_{3}⟩, ⟨b_{1}, b_{2}, b_{3}⟩, ⟨c_{1}, c_{2}, c_{3}⟩]$ $\overset{assign}{\to}$

Form the set of equations $U_{i} \cdot V_{j} = δ_{ij}$

•	Write $U_{i} \cdot V_{j} = δ (i, j)$

•	Context Panel: Sequence≻ $i$ ≻ (See Figure 1.5.11(a).)

•	Context Panel: Conversions≻To List

•	Context Panel: Sequence≻ $j$ ≻(See Figure 1.5.11(a).)

•	Context Panel: Join

•	Context Panel: Solve≻Solve

•	Context Panel: Assign to a Name≻ $S$

Figure 1.5.11(a) Sequence dialog

$U_{i} \cdot V_{j} = δ (i, j)$

${[[\begin{array}{c} 3 \\ −2 \\ 4 \end{array}], [\begin{array}{c} 2 \\ 5 \\ −4 \end{array}], [\begin{array}{c} 5 \\ 7 \\ 6 \end{array}]]}_{i} \cdot {[[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]]}_{j} = \{\begin{array}{c} 1 & i = j \\ 0 & i \neq j \end{array}$

$\overset{sequence w.r.t. i}{\to}$

$[\begin{array}{c} 3 \\ −2 \\ 4 \end{array}] \cdot {[[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]]}_{j} = \{\begin{array}{c} 1 & 1 = j \\ 0 & 1 \neq j \end{array}, [\begin{array}{c} 2 \\ 5 \\ −4 \end{array}] \cdot {[[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]]}_{j} = \{\begin{array}{c} 1 & 2 = j \\ 0 & 2 \neq j \end{array}, [\begin{array}{c} 5 \\ 7 \\ 6 \end{array}] \cdot {[[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]]}_{j} = \{\begin{array}{c} 1 & 3 = j \\ 0 & 3 \neq j \end{array}$

$\overset{to list}{\to}$

$[[\begin{array}{c} 3 \\ −2 \\ 4 \end{array}] \cdot {[[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]]}_{j} = \{\begin{array}{c} 1 & 1 = j \\ 0 & 1 \neq j \end{array}, [\begin{array}{c} 2 \\ 5 \\ −4 \end{array}] \cdot {[[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]]}_{j} = \{\begin{array}{c} 1 & 2 = j \\ 0 & 2 \neq j \end{array}, [\begin{array}{c} 5 \\ 7 \\ 6 \end{array}] \cdot {[[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \end{array}], [\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}], [\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}]]}_{j} = \{\begin{array}{c} 1 & 3 = j \\ 0 & 3 \neq j \end{array}]$

$\overset{sequence w.r.t. j}{\to}$

$[3 a_{1} - 2 a_{2} + 4 a_{3} = 1, 2 a_{1} + 5 a_{2} - 4 a_{3} = 0, 5 a_{1} + 7 a_{2} + 6 a_{3} = 0], [3 b_{1} - 2 b_{2} + 4 b_{3} = 0, 2 b_{1} + 5 b_{2} - 4 b_{3} = 1, 5 b_{1} + 7 b_{2} + 6 b_{3} = 0], [3 c_{1} - 2 c_{2} + 4 c_{3} = 0, 2 c_{1} + 5 c_{2} - 4 c_{3} = 0, 5 c_{1} + 7 c_{2} + 6 c_{3} = 1]$

$\overset{list join}{\to}$

$[3 a_{1} - 2 a_{2} + 4 a_{3} = 1, 2 a_{1} + 5 a_{2} - 4 a_{3} = 0, 5 a_{1} + 7 a_{2} + 6 a_{3} = 0, 3 b_{1} - 2 b_{2} + 4 b_{3} = 0, 2 b_{1} + 5 b_{2} - 4 b_{3} = 1, 5 b_{1} + 7 b_{2} + 6 b_{3} = 0, 3 c_{1} - 2 c_{2} + 4 c_{3} = 0, 2 c_{1} + 5 c_{2} - 4 c_{3} = 0, 5 c_{1} + 7 c_{2} + 6 c_{3} = 1]$

$\overset{solve}{\to}$

$\{a_{1} = \frac{29}{97}, a_{2} = - \frac{16}{97}, a_{3} = - \frac{11}{194}, b_{1} = \frac{20}{97}, b_{2} = - \frac{1}{97}, b_{3} = - \frac{31}{194}, c_{1} = - \frac{6}{97}, c_{2} = \frac{10}{97}, c_{3} = \frac{19}{194}\}$

$\overset{assign to a name}{\to}$

$S$

Evaluate each $V_{k}$ at the solution $S$

•	Expression palette: Evaluation template

•	Context Panel: Evaluate and Display Inline

$\binom{V_{1}}{} | \binom{}{S}$ = $[\begin{array}{c} \frac{29}{97} \\ - \frac{16}{97} \\ - \frac{11}{194} \end{array}]$

$\binom{V_{2}}{} | \binom{}{S}$ = $[\begin{array}{c} \frac{20}{97} \\ - \frac{1}{97} \\ - \frac{31}{194} \end{array}]$

$\binom{V_{3}}{} | \binom{}{S}$ = $[\begin{array}{c} - \frac{6}{97} \\ \frac{10}{97} \\ \frac{19}{194} \end{array}]$

The Context Panel option "Sequence" applies the seq command to generate a sequence of objects of similar structure. It cannot again be applied a second time to the resulting sequence of objects. Hence, the original sequence must be converted to a list, whereupon the Sequence option can again be applied. This results in a sequence of lists that can be combined into a single list via the Join option .

Maple Solution - Coded

•	Install the Student MultivariateCalculus package.

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

•	Use the piecewise command to define the function

$δ (i, j) = {\begin{array}{c} 1 & i = j \\ 0 & i \neq j \end{array}$

$δ ≔ (i, j) \to piecewise (i = j, 1, i \neq j, 0) &colon;$

•	Let U be the list of vectors A, B, and C.

$U ≔ [⟨3, - 2, 4⟩, ⟨2, 5, - 4⟩, ⟨5, 7, 6⟩] &colon;$

•	Let V be the list of vectors $V_{1}$ , $V_{2}$ , $V_{3}$ .

$V ≔ [⟨a_{1}, a_{2}, a_{3}⟩, ⟨b_{1}, b_{2}, b_{3}⟩, ⟨c_{1}, c_{2}, c_{3}⟩] &colon;$

•	Use the seq command to generate the equations $U_{i} \cdot V_{j} = δ_{ij}$ .

•	Apply the DotProduct command and assign the sequence of equations to the name $q$ .

$q ≔ seq (seq (DotProduct (U_{i}, V_{j}) = δ (i, j), i = 1 .. 3), j = 1 .. 3)$

$q ≔ 3 a_{1} - 2 a_{2} + 4 a_{3} = 1, 2 a_{1} + 5 a_{2} - 4 a_{3} = 0, 5 a_{1} + 7 a_{2} + 6 a_{3} = 0, 3 b_{1} - 2 b_{2} + 4 b_{3} = 0, 2 b_{1} + 5 b_{2} - 4 b_{3} = 1, 5 b_{1} + 7 b_{2} + 6 b_{3} = 0, 3 c_{1} - 2 c_{2} + 4 c_{3} = 0, 2 c_{1} + 5 c_{2} - 4 c_{3} = 0, 5 c_{1} + 7 c_{2} + 6 c_{3} = 1$

•	Apply the solve command to the set of equations $U_{i} \cdot V_{j} = δ_{ij}$

•	Assign the solution to the name $S$ .

$S ≔ solve (\{q\})$

$S ≔ \{a_{1} = \frac{29}{97}, a_{2} = - \frac{16}{97}, a_{3} = - \frac{11}{194}, b_{1} = \frac{20}{97}, b_{2} = - \frac{1}{97}, b_{3} = - \frac{31}{194}, c_{1} = - \frac{6}{97}, c_{2} = \frac{10}{97}, c_{3} = \frac{19}{194}\}$

•	Use the eval command to evaluate each vector in the list V with the values in the set $S$ .

$eval (V_{1}, S)$ = $[\begin{array}{c} \frac{29}{97} \\ - \frac{16}{97} \\ - \frac{11}{194} \end{array}]$

$eval (V_{2}, S)$ = $[\begin{array}{c} \frac{20}{97} \\ - \frac{1}{97} \\ - \frac{31}{194} \end{array}]$

$eval (V_{3}, S)$ = $[\begin{array}{c} - \frac{6}{97} \\ \frac{10}{97} \\ \frac{19}{194} \end{array}]$

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