To find the length of AB, where A is the foot of the perpendicular from the origin to the plane \( x - 2y + 2z + 6 = 0 \) and B is the point \( (0, -1, -4) \), we will follow these steps:
### Step 1: Identify the Plane and the Point
The equation of the plane is given as:
\[
x - 2y + 2z + 6 = 0
\]
The point B is given as:
\[
B(0, -1, -4)
\]
### Step 2: Find the Foot of the Perpendicular (Point A)
To find the foot of the perpendicular from the origin (0, 0, 0) to the plane, we can use the formula for the coordinates of the foot of the perpendicular from a point \( (x_1, y_1, z_1) \) to the plane \( ax + by + cz + d = 0 \):
\[
\left( x_1 - \frac{a(ax_1 + by_1 + cz_1 + d)}{a^2 + b^2 + c^2}, y_1 - \frac{b(ax_1 + by_1 + cz_1 + d)}{a^2 + b^2 + c^2}, z_1 - \frac{c(ax_1 + by_1 + cz_1 + d)}{a^2 + b^2 + c^2} \right)
\]
Here, \( a = 1, b = -2, c = 2, d = 6 \) and the point is \( (0, 0, 0) \).
### Step 3: Calculate the Values
1. Calculate \( ax_1 + by_1 + cz_1 + d \):
\[
1 \cdot 0 + (-2) \cdot 0 + 2 \cdot 0 + 6 = 6
\]
2. Calculate \( a^2 + b^2 + c^2 \):
\[
1^2 + (-2)^2 + 2^2 = 1 + 4 + 4 = 9
\]
3. Substitute these values into the formula for A:
\[
A = \left( 0 - \frac{1 \cdot 6}{9}, 0 - \frac{-2 \cdot 6}{9}, 0 - \frac{2 \cdot 6}{9} \right)
\]
\[
A = \left( -\frac{2}{3}, \frac{4}{3}, -\frac{4}{3} \right)
\]
### Step 4: Calculate the Length of AB
Now, we can find the distance \( AB \) using the distance formula:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
Where \( A = \left( -\frac{2}{3}, \frac{4}{3}, -\frac{4}{3} \right) \) and \( B = (0, -1, -4) \).
1. Calculate \( x_2 - x_1 \):
\[
0 - \left(-\frac{2}{3}\right) = \frac{2}{3}
\]
2. Calculate \( y_2 - y_1 \):
\[
-1 - \frac{4}{3} = -\frac{3}{3} - \frac{4}{3} = -\frac{7}{3}
\]
3. Calculate \( z_2 - z_1 \):
\[
-4 - \left(-\frac{4}{3}\right) = -4 + \frac{4}{3} = -\frac{12}{3} + \frac{4}{3} = -\frac{8}{3}
\]
4. Substitute these into the distance formula:
\[
AB = \sqrt{\left(\frac{2}{3}\right)^2 + \left(-\frac{7}{3}\right)^2 + \left(-\frac{8}{3}\right)^2}
\]
\[
= \sqrt{\frac{4}{9} + \frac{49}{9} + \frac{64}{9}} = \sqrt{\frac{117}{9}} = \sqrt{13}
\]
### Final Answer
The length of AB is:
\[
\sqrt{13}
\]
