Q.L is the foot of the perpendicular drawn from a point $P(3,4,5)$ on the $xy$-plane. The coordinates of point L are
(A) $(3,0,0)$
(B) $(0,4,5)$
(C) $(3,0,5)$
(D) none of these

The foot of the perpendicular from a point to a coordinate plane is found by dropping the coordinate perpendicular to that plane.

The $xy$-plane has equation $z = 0$ (all points with zero $z$-coordinate). When we drop a perpendicular from $P(3, 4, 5)$ to this plane, the perpendicular is vertical (parallel to the $z$-axis), so the $x$ and $y$ coordinates remain unchanged while $z$ becomes $0$. …

Understanding the Coordinate Planes

When we talk about the $xy$-plane in three-dimensional space, we're referring to the set of all points where the $z$-coordinate is zero. Think of it as the "floor" of our 3D coordinate system. Every point on this plane has the form $(x, y, 0)$.

Similarly:

• The $xz$-plane consists of points $(x, 0, z)$ (where $y = 0$)
• The $yz$-plane consists of points $(0, y, z)$ (where $x = 0$)

When we drop a perpendicular from a point to a plane, we're finding the shortest distance to that plane. The key insight is that a perpendicular to the $xy$-plane must be parallel to the $z$-axis—it moves straight up or down without any horizontal displacement.

Finding the Foot of the Perpendicular

1. Identify what stays constant

   Since the perpendicular to the $xy$-plane is vertical (parallel to the $z$-axis), the point $L$ must lie directly below (or above) $P$ in the $z$-direction. This means the $x$ and $y$ coordinates remain unchanged.

   From $P(3, 4, 5)$: the $x$-coordinate is $3$ and the $y$-coordinate is $4$.

2. Apply the plane constraint

   Point $L$ must lie on the $xy$-plane, so its $z$-coordinate must satisfy the plane's equation $z = 0$.

3. Combine the conditions

   The foot of the perpendicular is:

4. Check against the options
   • Option (A): $(3, 0, 0)$ — incorrect, the $y$-coordinate changed …