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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q31E (page 19)

Balancing a chemical reaction. Consider the chemical reaction
$a {NO}_{2} + b H_{2} O 鈫� c {HNO}_{2} + d {HNO}_{3}$ ,
where a, b, c, and d are unknown positive integers. The reaction must be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same,
$2 a + b = 2 c + 3 d$ .
While there are many possible values for $a, b, c$ and d that balance the reaction, it is customary to use the smallest possible positive integers. Balance this reaction.

Short Answer

Expert verified

The smallest possible positive integers are. $a = 2, b = c = d = 1$ ,

Step by step solution

01

Write the elements in terms of positive integers.

The elements in terms of integers are:

Oxygen:

$2 a + b = 2 c a + 3 d 2 a + b - 2 c - 3 d = 0 . . . . . . (1)$

Nitrogen:

$a = c + d a - c - d = 0 . . . . . . (2)$

Hydrogen:

$2 b = c + d 2 b - c - d = 0 . . . . . . (3)$

02

Consider the equations.

Re-write the equations in terms of a matrix.

$|\begin{matrix} 2 a + b - 2 c - 3 d = 0 \\ a - c - d = 0 \\ 2 b - c - d = 0 \end{matrix}|$

Represent the above system of equations in terms of the matrix form,

$[\begin{matrix} 2 & 1 & - 2 & - 3 & 0 \\ 1 & 0 & - 1 & - 1 & 0 \\ 0 & 2 & - 1 & - 1 & 0 \end{matrix}]$

03

Solve the matrix.

Consider the matrix.

$[\begin{matrix} 2 & 1 & - 2 & - 3 & 0 \\ 1 & 0 & - 1 & - 1 & 0 \\ 0 & 2 & - 1 & - 1 & 0 \end{matrix}]$

Using row Echelon form to reduce the matrix.

$[\begin{matrix} 2 & 1 & - 2 & - 3 & 0 \\ 0 & 2 & - 1 & - 1 & 0 \\ 0 & 0 & - \frac{1}{4} & \frac{1}{4} & 0 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & - 2 & 0 \\ 0 & 1 & 0 & - 1 & 0 \\ 0 & 0 & 1 & - 1 & 0 \end{matrix}]$

Consider the equations,

$a - 2 d = 0 . . . . . . . . (1) b - c = 0 . . . . . . . . (2) c - d = 0 . . . . . . . . . (3)$

Computing the equations (1), (2) and (3), we get the values as, $a = 2, b = c = d = 1$ .

04

Final answer

$a = 2, b = c = d = 1$ are the smallest possible values.

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Most popular questions from this chapter

In Exercises19through 24 , find the matrix Bof the linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ with respect to the basis $I = (\overset{鈬赌}{v_{1}}, \overset{鈬赌}{v_{2}})$ . For practice, solve each problem in three ways: (a) Use the formula $B = S^{- 1} A S$ , (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct 鈥渃olumn by column.鈥�

localid="1664194720187" $A = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}); \overset{鈬赌}{v_{1}} = [\begin{matrix} 1 \\ 1 \end{matrix}]; \overset{鈬赌}{v_{2}} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$

Find all solutions $x_{1}, x_{2}, x_{3}$ of the equation

$\overset{鈫�}{b} = x_{1} \overset{鈫�}{v_{1}} + x_{2} \overset{鈫�}{v_{2}} + x_{3} {\overset{鈫�}{v}}_{3}$

Where

$\overset{鈫�}{b} = [\begin{matrix} - 8 \\ - 1 \\ 2 \\ 15 \end{matrix}], \overset{鈫�}{v_{1}} = [\begin{matrix} 1 \\ 4 \\ 7 \\ 5 \end{matrix}], \overset{鈫�}{v_{2}} = [\begin{matrix} 2 \\ 5 \\ 8 \\ 3 \end{matrix}], \overset{鈫�}{v_{3}} = [\begin{matrix} 4 \\ 6 \\ 9 \\ 1 \end{matrix}]$

Consider the subspace Wof $R^{4}$ spanned by the vectors ${\overset{鈬赌}{V}}_{1} = [\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}] and {\overset{鈬赌}{V}}_{1} = [\begin{matrix} 1 \\ 9 \\ - 5 \\ 3 \end{matrix}]$

and Find the matrix of the orthogonal projection onto W.

Consider some particles in the plane with position vectors $\overset{鈫�}{r_{1}}, \overset{鈫�}{r_{2}}, . ...., \overset{鈫�}{r_{n}}$ and masses $m_{1}, m_{2}, . ...., m_{n}$ .

The position vector of the center of mass of this system is

$\overset{鈫�}{r_{c m}} = \frac{1}{M} (m_{1} \overset{鈫�}{r_{1}} + m_{2} \overset{鈫�}{r_{2}} + . .... + m_{n} \overset{鈫�}{r_{n}})$

where $M = m_{1} + m_{2} + . .... + m_{n}$ .

Consider the triangular plate shown in the accompanying sketch. How must a total mass of be distributed among the three vertices of the plate so that the plate can be supported at the point $[\begin{matrix} 2 \\ 2 \end{matrix}]$ ; that is, $\overset{鈫�}{r_{c m}} = [\begin{matrix} 2 \\ 2 \end{matrix}]$ ? Assume that the mass of the plate itself is negligible.

Let $S (t)$ be the length of the $t^{t h}$ day of the year $2013$ in Mumbai (formerly known as Bombay), India (measured in hours, from sunrise to sunset). We are given the following values of $S (t)$ :

$[\begin{matrix} t & S (t) \\ 47 & 11 .5 \\ 74 & 12 \\ 273 & 12 \end{matrix}]$

For example $S (47) = 11 .5$ , means that the timefrom sunrise to sunset on February16is11hours and30minutes. For locations close to the equator, the function $S (t)$ is well approximated by a trigonometric functionof the form

$S (t) = a + b c o s (\frac{2 蟺 t}{365}) + c s i n (\frac{2 蟺 t}{365})$

(The period is 365 days, or 1 year.) Find this approximationfor Mumbai, and graph your solution. Accordingto this model, how long is the longest day of the year inMumbai?

See all solutions

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