Q.The reflection of the point $(\alpha, \beta, \gamma)$ in the $xy$-plane is
(A) $(\alpha, \beta, 0)$
(B) $(0, 0, \gamma)$
(C) $(-\alpha, -\beta, \gamma)$
(D) $(\alpha, \beta, -\gamma)$

The key idea is that reflecting in the $xy$-plane keeps the $x$ and $y$ coordinates unchanged, while the $z$-coordinate changes sign (the plane acts as a mirror at $z=0$).

• The $xy$-plane is defined by $z=0$.
• A point $(\alpha, \beta, \gamma)$ is at a perpendicular distance $|\gamma|$ from this plane.
• Its mirror image will be the same distance on the opposite side, so the $z$-coordinate becomes $-\gamma$. …

The key idea here is understanding what "reflection in the $xy$-plane" actually means. The $xy$-plane is the plane where $z = 0$ — it's the flat horizontal surface that contains the $x$ and $y$ axes. When you reflect a point across this plane, you're essentially flipping it over that flat surface, like a mirror placed on the floor.

Think about it physically: if you stand on a glass floor and look down, your reflection appears exactly below you at the same distance. Your horizontal position (left-right and forward-backward) doesn't change — only your vertical position flips. That's exactly what happens here: the $x$ and $y$ coordinates stay the same, but the $z$ coordinate changes sign.

Let's work through it step by step.

1. Identify the reflecting surface. The $xy$-plane is defined by $z = 0$. Any point in space has coordinates $(x, y, z)$. The plane acts as a mirror: the perpendicular distance from the point to the plane is simply $|z|$, since the plane is horizontal.

2. Understand what reflection does. Reflection across a plane sends a point to the opposite side of the plane, at the same perpendicular distance. If the original point is at height $z$ above the plane (positive $z$), its reflection will be at height $z$ below the plane (negative $z$). If the original is below the plane (negative $z$), the reflection goes above (positive $z$). In both cases, the new $z$-coordinate is $-z$. …

Method: Reflecting a Point in a Coordinate Plane

Reflection acts like a mirror: the coordinate perpendicular to the mirror flips sign, the other two are untouched.

Steps

Step 1: Find the coordinate perpendicular to the mirror.

The $xy$-plane is $z=0$, so the perpendicular direction is $z$. (For the $yz$-plane it is $x$; for the $zx$-plane it is $y$.)

Step 2: Flip the sign of that one coordinate; leave the rest.

Reflecting $(\alpha,\beta,\gamma)$ in the $xy$-plane sends $\gamma\to-\gamma$ and keeps $\alpha,\beta$:

$(\alpha,\beta,\gamma)\ \longmapsto\ (\alpha,\beta,-\gamma).$ …

Common Mistakes

Mistake 1: Confusing reflection with projection.

Why it's wrong: setting $z=0$ (giving $(\alpha,\beta,0)$) is the projection onto the $xy$-plane; a reflection sends the point to the opposite side, so $z\to-\gamma$. Correct approach: flip the sign of the perpendicular coordinate rather than zeroing it.

Mistake 2: Flipping the wrong coordinates. …