(a) Find the Cartesian equation of the line ℓ through Q in direction v.

The equation of a line in 2D space is given by y = mx + c, where m is the slope of the line and c is the y-intercept.

The slope of the line ℓ is given by the ratio of the y-component to the x-component of the vector v, which is 2/3.

The line passes through the point Q = (2, 3), so we can substitute these values into the equation to find the y-intercept:

3 = (2/3)*2 + c
3 = 4/3 + c
c = 3 - 4/3 = 5/3

So, the Cartesian equation of the line ℓ is y = (2/3)x + 5/3.

(b) Find the Cartesian equation of the line m such that ρQ,π/2 = σℓσm.

The notation ρQ,π/2 represents a rotation by π/2 radians (or 90 degrees) about the point Q, and σℓσm represents the reflection of the line ℓ in the line m.

A line that is rotated by 90 degrees about a point has a slope that is the negative reciprocal of the original line's slope. So, the slope of the line m is -3/2.

The line m also passes through the point Q = (2, 3), so we can substitute these values into the equation to find the y-intercept:

3 = (-3/2)*2 + c
3 = -3 + c
c = 6

So, the Cartesian equation of the line m is y = (-3/2)x + 6.

(c) What is the isometry σ−1m ◦ ρQ,π/2 ◦ σm?

The notation σ−1m represents the reflection in the line m, ρQ,π/2 represents a rotation by π/2 radians about the point Q, and σm represents the reflection in the line m.

The composition of these transformations, σ−1m ◦ ρQ,π/2 ◦ σm, represents a reflection in the line m, followed by a rotation about the point Q, followed by another reflection in the line m.

This is a complex transformation that cannot be easily described in terms of simple geometric transformations. However, it is an isometry, which means it preserves distances and angles, so it can be thought of as a combination of rotations, reflections, and translations.

Question

(a) Find the Cartesian equation of the line ℓ through Q in direction v.

The equation of a line in 2D space is given by y = mx + c, where m is the slope of the line and c is the y-intercept.

The slope of the line ℓ is given by the ratio of the y-component to the x-component of the vector v, which is 2/3.

The line passes through the point Q = (2, 3), so we can substitute these values into the equation to find the y-intercept:

3 = (2/3)*2 + c
3 = 4/3 + c
c = 3 - 4/3 = 5/3

So, the Cartesian equation of the line ℓ is y = (2/3)x + 5/3.

(b) Find the Cartesian equation of the line m such that ρQ,π/2 = σℓσm.

The notation ρQ,π/2 represents a rotation by π/2 radians (or 90 degrees) about the point Q, and σℓσm represents the reflection of the line ℓ in the line m.

A line that is rotated by 90 degrees about a point has a slope that is the negative reciprocal of the original line's slope. So, the slope of the line m is -3/2.

The line m also passes through the point Q = (2, 3), so we can substitute these values into the equation to find the y-intercept:

3 = (-3/2)*2 + c
3 = -3 + c
c = 6

So, the Cartesian equation of the line m is y = (-3/2)x + 6.

(c) What is the isometry σ−1m ◦ ρQ,π/2 ◦ σm?

The notation σ−1m represents the reflection in the line m, ρQ,π/2 represents a rotation by π/2 radians about the point Q, and σm represents the reflection in the line m.

The composition of these transformations, σ−1m ◦ ρQ,π/2 ◦ σm, represents a reflection in the line m, followed by a rotation about the point Q, followed by another reflection in the line m.

This is a complex transformation that cannot be easily described in terms of simple geometric transformations. However, it is an isometry, which means it preserves distances and angles, so it can be thought of as a combination of rotations, reflections, and translations.

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