Find the coordinates of the foot of perpendicular drawn from origin to the planes: `x+y+z=1`

Answer

To find the coordinates of the foot of the perpendicular drawn from the origin to the plane given by the equation \(x + y + z = 1\), we can follow these steps:

### Step 1: Identify the Plane Equation
The equation of the plane can be rewritten in the standard form:
\[ x + y + z - 1 = 0 \]
Here, we can identify the coefficients:
- \(a = 1\)
- \(b = 1\)
- \(c = 1\)
- \(d = -1\)

### Step 2: Use the Foot of Perpendicular Formula
The formula for finding the foot of the perpendicular from a point \((x_1, y_1, z_1)\) to the plane \(ax + by + cz + d = 0\) is given by:
\[ \left( x_1 - \frac{a(ax_1 + by_1 + cz_1 + d)}{a^2 + b^2 + c^2}, y_1 - \frac{b(ax_1 + by_1 + cz_1 + d)}{a^2 + b^2 + c^2}, z_1 - \frac{c(ax_1 + by_1 + cz_1 + d)}{a^2 + b^2 + c^2} \right) \]

### Step 3: Substitute the Origin Coordinates
Since we are drawing the perpendicular from the origin, we have:
\[ (x_1, y_1, z_1) = (0, 0, 0) \]
Substituting these values into the formula gives:
\[ \left( 0 - \frac{a(0 + 0 + 0 - 1)}{a^2 + b^2 + c^2}, 0 - \frac{b(0 + 0 + 0 - 1)}{a^2 + b^2 + c^2}, 0 - \frac{c(0 + 0 + 0 - 1)}{a^2 + b^2 + c^2} \right) \]

### Step 4: Calculate the Denominator
Now, calculate \(a^2 + b^2 + c^2\):
\[ a^2 + b^2 + c^2 = 1^2 + 1^2 + 1^2 = 3 \]

### Step 5: Calculate the Numerator
Next, calculate \(ax_1 + by_1 + cz_1 + d\):
\[ 1(0) + 1(0) + 1(0) - 1 = -1 \]

### Step 6: Substitute Values into the Formula
Now substituting back into the formula:
\[ \left( 0 - \frac{1 \cdot (-1)}{3}, 0 - \frac{1 \cdot (-1)}{3}, 0 - \frac{1 \cdot (-1)}{3} \right) = \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right) \]

### Conclusion
Thus, the coordinates of the foot of the perpendicular from the origin to the plane \(x + y + z = 1\) are:
\[ \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right) \]