Q41E In Exercise 40 through 46, consi... [FREE SOLUTION]

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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 5: Q41E (page 217)

In Exercise 40 through 46, consider vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}$ in $R^{4}$ ; we are told that ${\overset{鈫�}{v}}_{i}, {\overset{鈫�}{v}}_{j}$ is the entry $a_{i j}$ of matrix A.
$A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}]$
Find the angle enclosed by vectors ${\overset{鈫�}{v}}_{2}$ and ${\overset{鈫�}{v}}_{3}$

Short Answer

Expert verified

The required angle is $c o s^{- 1} \frac{20}{21} 鈮� 0.31$ .

Step by step solution

Step by step solutionStep 1: Given

The given matrix is $A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}]$

Angle between v→2 and v→3

To find the angle between ${\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}$ . Let us write the matrix in the ${\overset{鈫�}{v}}_{i} 路 {\overset{鈫�}{v}}_{j}$ notation.

Consider the terms below:

$\begin{array}{rcl} A & = & [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}] \\ = & [\begin{matrix} {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{2} & {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{2} & {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{3} \end{matrix}] \\ = & [\begin{matrix} | | {\overset{鈫�}{v}}_{1} | |^{2} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{1} & | | {\overset{鈫�}{v}}_{2} | |^{2} & {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{2} & | | {\overset{鈫�}{v}}_{3} | |^{2} \end{matrix}] \end{array}$

Now, let the angle be $胃_{23}$ .

Observe that to find out the value of role="math" localid="1659605838938" $胃_{23}$ :

$\begin{array}{rcl} {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{3} & = & | | {\overset{鈫�}{v}}_{2} | | | | {\overset{鈫�}{v}}_{3} | | \cos 胃_{23} \\ 20 & = & \sqrt{9} 脳 \sqrt{49} \cos 胃_{23} \\ \cos 胃_{23} & = & \frac{20}{21} \end{array}$

Simplify as:

role="math" localid="1659606029016" $\begin{array}{rcl} 胃_{23} & = & \cos^{- 1} \frac{20}{21} \\ 鈮� & 0.31 \end{array}$

Hence, the angle $胃_{23}$ is $0.31$ radians.

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91影视

Short Answer

Step by step solution

Step by step solutionStep 1: Given

Angle between v→2 and v→3

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91影视

In Exercise 40 through 46, consider vectors v鈫�1,v鈫�2,v鈫�3in R4; we are told that v鈫�i,v鈫�jis the entry aij of matrix A.A=[35115920112049]Find the angle enclosed by vectors v鈫�2and v鈫�3

Short Answer

Step by step solution

Step by step solutionStep 1: Given

Angle between v→2 and v→3

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