All of the points on triangle ABC undergo the same change to form DEF. Triangle DEF is formed by reflecting ABC across the y-axis and has vertices D (4, -6), E (6, -2) and F (2, -4). y-axis reflectionĪ reflection across the y-axis changes the position of the x-coordinate of all the points in a figure such that (x, y) becomes (-x, y). Triangle DEF is formed by reflecting ABC across the x-axis and has vertices D (-6, -2), E (-4, -6) and F (-2, -4). x-axis reflectionĪ reflection across the x-axis changes the position of the y-coordinate of all the points in a figure such that (x, y) becomes (x, -y). Reflections in coordinate geometryīelow are three examples of reflections in coordinate plane. This is true for the distances between any corresponding points and the line of reflection, so line l is also a line of symmetry. Points A, B, and C on the pentagon are reflected across line l to A', B', and C'. Let line l be a line of reflection for the pentagon above. Whenever you reflect a figure across a line of reflection that is also a line of symmetry, each point on the figure is translated an equal distance across the line of symmetry, back on to the figure. You can think of folding half of the image of the butterfly across the line of reflection back on to its other half. The same result occurs if the left side of the butterfly is reflected across line l, so line l is also a line of symmetry. Reflecting the right side of the butterfly across line l maps it to the butterfly's left side. Reflection symmetryĪ line of reflection is also a line of symmetry if a geometric shape or figure can be reflected across the line back onto itself. This is true for any corresponding points on the two triangles. A, B, and C are the same distance from the line of reflection as their corresponding points, D, E, and F. In the figure above, triangle ABC is reflected across the line to form triangle DEF. For a 3D object, each point moves the same distance across a plane of refection. In a reflection of a 2D object, each point on the preimage moves the same distance across a line, called the line of reflection, to form a mirror image of itself. The term "preimage" is used to describe a geometric figure before it has been transformed and the term "image" is used to describe it after it has been transformed. A reflection is a rigid transformation, which means that the size and shape of the figure does not change the figures are congruent before and after the transformation. In geometry, a reflection is a type of transformation in which a shape or geometric figure is mirrored across a line or plane. Then consider how you think reflections would work in $\Bbb R^n$ for other values of $n$.Home / geometry / transformation / reflection Reflection Try to visualize each of these reflections in $\Bbb R^2$ and $\Bbb R^3$. Go back and look up the geometric properties of even and odd functions if you don't remember how these reflections work in $\Bbb R^2$ (note however that you can still reflect through the origin in $\Bbb R^3$). The difference between reflecting through a line vs a plane in $\Bbb R^3$ is comparable to reflecting through the origin vs a line in $\Bbb R^2$.
![reflect across y axis reflect across y axis](https://i.ytimg.com/vi/eSMZeC2qJQs/maxresdefault.jpg)
Let's see how this affects the standard basis $\$$
![reflect across y axis reflect across y axis](https://us-static.z-dn.net/files/dc6/b79c1a22335ab670f0c78e269003ffbe.png)
![reflect across y axis reflect across y axis](https://i.ytimg.com/vi/Ut-Z_KhMhDI/maxresdefault.jpg)
The far end of that line segment is then at the point that is the reflection of your point across the $y$-axis. Extend that line segment past $y$ by the same length as the distance from the point to the $y$-axis. Now connect that point to the $y$-axis by a line segment that is orthogonal to the $y$-axis. Consider an arbitrary point in $\Bbb R^3$. Unfortunately I can't find a good image on Google Images to describe reflection through a line in $\Bbb R^3$ (and my pgfplots-fu is still pretty basic), but I'll try to describe what it means.
![reflect across y axis reflect across y axis](https://us-static.z-dn.net/files/d4f/3a64614c26771f39c9494b457181a84f.png)
When a question asks you to find a matrix representing a linear transformation $T$ that is only described geometrically, your task is to figure out how that $T$ transforms a basis for your domain.