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Chapter 4: Q38E (page 191)

Exercises 37 and 38 concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying

figure. The vectors\(\left( {\begin{array}{*{20}{c}}{2.6}\\{ - 1.5}\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}0\\3\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}0\\0\\{4.8}\end{array}} \right)\)in\({\mathbb{R}^{\bf{3}}}\)form a basis for the unit cell shown on the right. The numbers here are Angstrom units\(\left( {1\mathop { A}\limits^{{\rm{ o}}} = 1{0^{ - 8}}cm} \right)\). In alloys of titanium, some additional atoms may be in the unit cell at the octahedral and tetrahedralsites (so named because of the geometric objects

formed by atoms at these locations).

The hexagonal close-packed lattice and its unit cell.

38. One of the tetrahedral sites is\(\left( {\begin{array}{*{20}{c}}{1/2}\\{1/{\bf{2}}}\\{1/{\bf{3}}}\end{array}} \right)\). Determine the coordinates of this site relative to the standard basis of\({\mathbb{R}^{\bf{3}}}\).

Short Answer

Expert verified

The coordinates of this site relative to the standard basis of \({\mathbb{R}^{\bf{3}}}\) is \({\bf{x}} = \left( {\begin{array}{*{20}{c}}{1.3}\\{0.75}\\{1.6}\end{array}} \right)\).

Step by step solution

01

Coordinates of x relative to the standard basis

It is given for the unit cell, thebasis is\(B = \left\{ {\left( {\begin{array}{*{20}{c}}{2.6}\\{ - 1.5}\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\3\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\0\\{4.8}\end{array}} \right)} \right\}\).

Obtain thecoordinates of the octahedral site\({\left( {\bf{x}} \right)_B} = \left( {\begin{array}{*{20}{c}}{1/2}\\{1/2}\\{1/3}\end{array}} \right)\)relative to the standardbasis of\({\mathbb{R}^{\bf{3}}}\)as shown below:

Write basis B in the matrix form as shown below:

\({P_B} = \left( {\begin{array}{*{20}{c}}{2.6}&0&0\\{ - 1.5}&3&0\\0&0&{4.8}\end{array}} \right)\)

02

State the coordinates of x relative to the standard basis

Compute the product of matrices\({\left( {\bf{x}} \right)_B} = \left( {\begin{array}{*{20}{c}}{1/2}\\{1/2}\\{1/3}\end{array}} \right)\)and\({P_B} = \left( {\begin{array}{*{20}{c}}{2.6}&0&0\\{ - 1.5}&3&0\\0&0&{4.8}\end{array}} \right)\)using the MATLAB command shown below:

\(\begin{array}{l} > > {\rm{A}} = \left( {{\rm{2}}{\rm{.6 0 0; }} - {\rm{1}}{\rm{.5 3 0; 0 0 4}}{\rm{.8}}} \right);\\ > > {\rm{B}} = \left( {{\rm{1/2 1/2 1/3}}} \right);\\ > > {\rm{M}} = {\rm{A}}*{\rm{B}}\end{array}\)

So, the output is

\(\begin{array}{c}{\bf{x}} = {P_B}{\left( {\bf{x}} \right)_B}\\ = \left( {\begin{array}{*{20}{c}}{2.6}&0&0\\{ - 1.5}&3&0\\0&0&{4.8}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{1/2}\\{1/2}\\{1/3}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{1.3}\\{0.75}\\{1.6}\end{array}} \right).\end{array}\)

Thus, the coordinates of this site relative to the standard basis of \({\mathbb{R}^{\bf{3}}}\) is \({\bf{x}} = \left( {\begin{array}{*{20}{c}}{1.3}\\{0.75}\\{1.6}\end{array}} \right)\).

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Most popular questions from this chapter

In Exercise 17, Ais an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if Ais not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

Define by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\). For instance, if \({\mathop{\rm p}\nolimits} \left( t \right) = 3 + 5t + 7{t^2}\), then \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}3\\{15}\end{array}} \right)\).

  1. Show that \(T\) is a linear transformation. (Hint: For arbitrary polynomials p, q in \({{\mathop{\rm P}\nolimits} _2}\), compute \(T\left( {{\mathop{\rm p}\nolimits} + {\mathop{\rm q}\nolimits} } \right)\) and \(T\left( {c{\mathop{\rm p}\nolimits} } \right)\).)
  2. Find a polynomial p in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\), and describe the range of \(T\).

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Let \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Find \({\mathop{\rm v}\nolimits} \) in \({\mathbb{R}^3}\) such that \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&4\\2&{ - 6}&8\end{array}} \right] = {{\mathop{\rm uv}\nolimits} ^T}\) .
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