Q48E If a 2脳2 matrix聽A represents a... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q48E (page 74)

If a $2 脳 2$ matrix $A$ represents a rotation, find perpendicular unit vectors $\overset{鈫�}{v_{1}}$ and $\overset{鈫�}{v_{2}}$ in ${鈩�}^{2}$ such that the vectors $T (\overset{鈫�}{v_{1}})$ and $T (\overset{鈫�}{v_{2}})$ areperpendicular as well. See Exercise47.

Short Answer

Expert verified

The perpendicular vectors are $\overset{鈫�}{v_{1}} = (\begin{array}{l} 1 \\ 0 \end{array}), \overset{鈫�}{v_{2}} = (\begin{array}{l} 0 \\ 1 \end{array})$ .

Step by step solution

Compute the function

Consider the matrix.

$A = (\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix})$

The function is,

$\begin{array}{l} f (t) = (\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}) (\begin{array}{l} \cos t \\ \sin t \end{array}) 脳 (\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}) (\begin{array}{l} 鈭� \sin t \\ \cos t \end{array}) \\ 鈭� f (t) = 0 \end{array}$

Compute the vectors and transformations

Consider the vectors.

$\begin{array}{l} \overset{鈫�}{v_{1}} = (\begin{array}{l} \cos 0 \\ \sin 0 \end{array}) = (\begin{array}{l} 1 \\ 0 \end{array}) \\ \overset{鈫�}{v_{2}} = (\begin{array}{l} 鈭� \sin 0 \\ \cos 0 \end{array}) = (\begin{array}{l} 0 \\ 1 \end{array}) \end{array}$

Consider the linear transformations.

$\begin{array}{l} T (\overset{鈫�}{v_{1}}) = (\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}) (\begin{array}{l} 1 \\ 0 \end{array}) = (\begin{array}{l} \cos 胃 \\ \sin 胃 \end{array}) \\ T (\overset{鈫�}{v_{2}}) = (\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}) (\begin{array}{l} 0 \\ 1 \end{array}) = (\begin{array}{l} 鈭� \sin 胃 \\ \cos 胃 \end{array}) \end{array}$

Verify the condition.

$T (\overset{鈫�}{v_{1}}) 脳 T (\overset{鈫�}{v_{2}}) = (\begin{array}{l} \cos 胃 \\ \sin 胃 \end{array}) 脳 (\begin{array}{l} 鈭� \sin 胃 \\ \cos 胃 \end{array}) = 0$

Therefore, the vectors and their linear transformations are perpendicular.

Hence, the perpendicular vectors are . $\overset{鈫�}{v_{1}} = (\begin{array}{l} 1 \\ 0 \end{array}), \overset{鈫�}{v_{2}} = (\begin{array}{l} 0 \\ 1 \end{array})$

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91影视

If a $2 脳 2$ matrix $A$ represents a rotation, find perpendicular unit vectors $\overset{鈫�}{v_{1}}$ and $\overset{鈫�}{v_{2}}$ in ${鈩�}^{2}$ such that the vectors $T (\overset{鈫�}{v_{1}})$ and $T (\overset{鈫�}{v_{2}})$ areperpendicular as well. See Exercise47.

Short Answer

Step by step solution

Compute the function

Compute the vectors and transformations

One App. One Place for Learning.

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91影视

If a 2脳2 matrixA represents a rotation, find perpendicular unit vectors v1鈫�and v2鈫�in 鈩�2such that the vectorsT(v1鈫�)andT(v2鈫�) areperpendicular as well. See Exercise47.

Short Answer

Step by step solution

Compute the function

Compute the vectors and transformations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

If a $2 脳 2$ matrix $A$ represents a rotation, find perpendicular unit vectors $\overset{鈫�}{v_{1}}$ and $\overset{鈫�}{v_{2}}$ in ${鈩�}^{2}$ such that the vectors $T (\overset{鈫�}{v_{1}})$ and $T (\overset{鈫�}{v_{2}})$ areperpendicular as well. See Exercise47.