Which of the indicated maps 𝐅(x, y) define isometries of the Euclidean plane? (a) ( y

Which of the indicated maps 𝐅(x, y) define isometries of the...

Which of the indicated maps $\mathbf{F}(x, y)$ define isometries of the Euclidean plane? (a) $\left(\begin{array}{c}y \\ -x\end{array}\right)$, (b) $\left(\begin{array}{l}x-2 \\ y-1\end{array}\right)$ (c) $\left(\begin{array}{c}x-y+1 \\ x+2\end{array}\right)$. (d) $\frac{1}{\sqrt{2}}\left(\begin{array}{l}x+y-3 \\ x+y-2\end{array}\right)$, (e) $\frac{1}{5}\left(\begin{array}{c}3 x+4 y \\ -4 x+3 y+1\end{array}\right)$.

Which of the indicated maps $\mathbf{F}(x, y)$ define isometries of the Euclidean plane?
(a) $\left(\begin{array}{c}y \\ -x\end{array}\right)$,
(b) $\left(\begin{array}{l}x-2 \\ y-1\end{array}\right)$
(c) $\left(\begin{array}{c}x-y+1 \\ x+2\end{array}\right)$.
(d) $\frac{1}{\sqrt{2}}\left(\begin{array}{l}x+y-3 \\ x+y-2\end{array}\right)$,
(e) $\frac{1}{5}\left(\begin{array}{c}3 x+4 y \\ -4 x+3 y+1\end{array}\right)$.

Key Concepts

Affine Transformation Representing Isometries

Many isometries can be represented as affine transformations of the form F(x) = Ax + b, where A is an orthogonal matrix (ensuring the linear part preserves distances and angles) and b is a fixed translation vector. This decomposition captures the essential idea behind most isometric maps in the Euclidean plane.

Euclidean Isometry

A Euclidean isometry is a transformation of the Euclidean plane that preserves distances between points. This means that the lengths of segments and the angles between lines remain unchanged after the transformation. In the context of the plane, these isometries include translations, rotations, reflections, and glide reflections.

Orthogonal Transformation

An orthogonal transformation is a linear map whose representing matrix Q satisfies Q^T Q = I. This property guarantees that lengths and angles are preserved, making orthogonal transformations a key component in describing rotations and reflections, which are fundamental types of isometries in the Euclidean plane.

Translation

A translation is a specific type of isometry in which every point is shifted by the same fixed vector. Because the relative distances between points remain constant, translations preserve the geometric structure of figures, making them a basic example of distance-preserving transformations.

Transcript

00:01 In problem 46 we have the vector b equals 4i plus 3g for example this vector and we have another vector c which is perpendicular to b then for example c would be this direction or in this direction we want to find the vector that is in the same plane having projections in one in the direction of b and the direction of b and the direction of c as 1 and 2 then if we have vector a for example this vector we want to get its projection on b and equated to 1 and its projection on c and equated to 2 let's find this vector a first we get we should get c which is perpendicular to be if we assume that c equal c1 i plus c2 j then the dot product between c and b is 0 then 4 c1 plus 3 c2 equal 0 then we can get that c1 equals minus 3 c2 divided by 4 and we can get a unit vector in the direction of c we can get c head c had equal c divided by the norm of c c equal c1 i plus c2 j divided by the norm of c which is square root of c1 squared plus c2 squared c1 squared plus c2 square let's substitute by c1 equals minus 3 divided by 4 c2 then it equals minus 3 divided by 4 i plus j all multiplied by c2 divided by c2 divided by c2 divided by square root of 3 divided by 4 squared plus 1 multiplied by c2 we can remove c2 with c2 then we have in the direction of c we have ionid vector equals 4 divided by 5 multiplied by minus 3 divided by 4 i plus j it equals 1 divided by 5 multiplied by minus 3 i plus 4g this is ioned vector in the direction of c hat to find a we get 2 multiplied by c hat plus 1 multiplied by b hat let's get b hat b hat is just b divided by the norm of b the norm of b is 5 then it's 1 divided by 5 multiplied by 4i plus let's get a a equals 2c 2 divided by 5 multiplied by 3 i plus 4 j plus 1 multiplied by b 1 divided by 5 multiplied by 4i plus 3 j doing the sums it equals minus 6 divided by 5 plus 4 divided by 5 gives minus 2 divided by 5 i 4 multiplied by 2 divided by 5 plus 3 divided by 5 gives 8 divided by 5 j sorry this is not 8 it it's 2 multiplied by 4 8 plus 3 equals 11, 11 divided by 5.

04:59 This is one option for a.

05:02 Another option is to consider that c in the opposite direction.

05:08 Then we have a equals minus 2 divided by 5 minus 3i plus 4j, plus 1 divided by 5, 4i plus 3g...

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