Find the length of the foot of the perpendicular from the point (1,1,2) to the plane `vecr.(2hati-2hatj+4hatk)+5=0`

Answer

To find the length of the foot of the perpendicular from the point \( P(1, 1, 2) \) to the plane given by the equation \( \vec{r} \cdot (2\hat{i} - 2\hat{j} + 4\hat{k}) + 5 = 0 \), we can follow these steps:

### Step 1: Write the equation of the plane in Cartesian form
The equation of the plane can be expressed in the form:
\[ \vec{r} \cdot \vec{n} + d = 0 \]
where \( \vec{n} \) is the normal vector of the plane and \( d \) is a constant. Here, the normal vector \( \vec{n} = (2, -2, 4) \) and \( d = 5 \).

Thus, the equation of the plane can be rewritten as:
\[ 2x - 2y + 4z + 5 = 0 \]

### Step 2: Identify the coordinates of the point
The coordinates of the point \( P \) are given as \( (1, 1, 2) \).

### Step 3: Use the formula for the distance from a point to a plane
The distance \( D \) from a point \( (x_0, y_0, z_0) \) to the plane \( Ax + By + Cz + D = 0 \) is given by:
\[ D = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
In our case:
- \( A = 2 \)
- \( B = -2 \)
- \( C = 4 \)
- \( D = 5 \)
- \( (x_0, y_0, z_0) = (1, 1, 2) \)

### Step 4: Substitute the values into the formula
Substituting the values into the distance formula:
\[ D = \frac{|2(1) - 2(1) + 4(2) + 5|}{\sqrt{2^2 + (-2)^2 + 4^2}} \]

### Step 5: Calculate the numerator
Calculating the numerator:
\[ 2(1) - 2(1) + 4(2) + 5 = 2 - 2 + 8 + 5 = 13 \]
Thus, the absolute value is:
\[ |13| = 13 \]

### Step 6: Calculate the denominator
Calculating the denominator:
\[ \sqrt{2^2 + (-2)^2 + 4^2} = \sqrt{4 + 4 + 16} = \sqrt{24} \]

### Step 7: Final calculation of distance
Now substituting back into the formula:
\[ D = \frac{13}{\sqrt{24}} = \frac{13}{2\sqrt{6}} = \frac{13\sqrt{6}}{12} \]

### Conclusion
The length of the foot of the perpendicular from the point \( (1, 1, 2) \) to the plane is:
\[ \frac{13\sqrt{6}}{12} \]