Find the foot of perpendicular drawn from a point M(-2, 3, 6) on the coordinate planes.

Answer

To find the foot of the perpendicular drawn from the point M(-2, 3, 6) to the coordinate planes, we need to understand the concept of projection onto the coordinate planes. The three coordinate planes are:

1. The XY-plane (where z = 0)
2. The XZ-plane (where y = 0)
3. The YZ-plane (where x = 0)

We will find the foot of the perpendicular from the point M to each of these planes.

### Step 1: Find the foot of the perpendicular to the XY-plane

The XY-plane is defined by the equation z = 0. To find the foot of the perpendicular from point M(-2, 3, 6) to the XY-plane, we keep the x and y coordinates the same and set z to 0.

- The coordinates of the foot of the perpendicular to the XY-plane are:
\[ F_{XY} = (-2, 3, 0) \]

### Step 2: Find the foot of the perpendicular to the XZ-plane

The XZ-plane is defined by the equation y = 0. To find the foot of the perpendicular from point M(-2, 3, 6) to the XZ-plane, we keep the x and z coordinates the same and set y to 0.

- The coordinates of the foot of the perpendicular to the XZ-plane are:
\[ F_{XZ} = (-2, 0, 6) \]

### Step 3: Find the foot of the perpendicular to the YZ-plane

The YZ-plane is defined by the equation x = 0. To find the foot of the perpendicular from point M(-2, 3, 6) to the YZ-plane, we keep the y and z coordinates the same and set x to 0.

- The coordinates of the foot of the perpendicular to the YZ-plane are:
\[ F_{YZ} = (0, 3, 6) \]

### Summary of Results

- Foot of the perpendicular to the XY-plane: \( F_{XY} = (-2, 3, 0) \)
- Foot of the perpendicular to the XZ-plane: \( F_{XZ} = (-2, 0, 6) \)
- Foot of the perpendicular to the YZ-plane: \( F_{YZ} = (0, 3, 6) \)