Question

1.8 (a) Let $\theta$ be a nonzero angle and $\mathbf{b}$ a translation vector in the plane. Give a geometric\nconstruction for a point $P \in \mathbb{E}^2$ such that\n$\text{Rot}(O, \theta)(P) = \text{Trans}(-\mathbf{b})(P)$.\n[Hint: draw a picture, to find points $P, Q$ with $\mathbf{b} = \overrightarrow{QP}$ such that $O$ is on the\nperpendicular bisector of $PQ$ and $\angle POQ = \theta$.]\n(b) By solving linear equations, find $x, y$ such that\n$A \begin{pmatrix} x_1\\x_2 \end{pmatrix} + \begin{pmatrix} b_1\\b_2 \end{pmatrix} = \begin{pmatrix} x_1\\x_2 \end{pmatrix}$, where $A = \begin{pmatrix} \cos \theta & \sin \theta\\\sin \theta & -\cos \theta \end{pmatrix}$.\n(c) Express the motion $T : \mathbb{E}^2 \to \mathbb{E}^2$ defined in coordinates by $T(\mathbf{x}) = A\mathbf{x} + \mathbf{b}$ in the\nform $T = \text{Rot}(P, \theta)$.\n(d) Relate (a) and (b).

          1.8 (a) Let $\theta$ be a nonzero angle and $\mathbf{b}$ a translation vector in the plane. Give a geometric\nconstruction for a point $P \in \mathbb{E}^2$ such that\n$\text{Rot}(O, \theta)(P) = \text{Trans}(-\mathbf{b})(P)$.\n[Hint: draw a picture, to find points $P, Q$ with $\mathbf{b} = \overrightarrow{QP}$ such that $O$ is on the\nperpendicular bisector of $PQ$ and $\angle POQ = \theta$.]\n(b) By solving linear equations, find $x, y$ such that\n$A \begin{pmatrix} x_1\\x_2 \end{pmatrix} + \begin{pmatrix} b_1\\b_2 \end{pmatrix} = \begin{pmatrix} x_1\\x_2 \end{pmatrix}$, where $A = \begin{pmatrix} \cos \theta & \sin \theta\\\sin \theta & -\cos \theta \end{pmatrix}$.\n(c) Express the motion $T : \mathbb{E}^2 \to \mathbb{E}^2$ defined in coordinates by $T(\mathbf{x}) = A\mathbf{x} + \mathbf{b}$ in the\nform $T = \text{Rot}(P, \theta)$.\n(d) Relate (a) and (b).

1.8 (a) Let θ be a nonzero angle and 𝐛 a translation vector in the plane. Give a geometricfor a point P ∈𝔼^2 such thatRot(O, θ)(P) = Trans(-𝐛)(P).[Hint: draw a picture, to find points P, Q with 𝐛 = QP such that O is on thebisector of PQ and ∠ POQ = θ.](b) By solving linear equations, find x, y such thatA
< p m a t r i x >
+
< p m a t r i x >
=
< p m a t r i x >, where A =
< p m a t r i x >.(c) Express the motion T : 𝔼^2 →𝔼^2 defined in coordinates by T(𝐱) = A𝐱 + 𝐛 in theT = Rot(P, θ).(d) Relate (a) and (b).

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