The distance of a plane `3x + 4y - 5z = 60 ` from
origin

Options

A. `3/sqrt(50)`
B. `6/sqrt(50)`
C. `6sqrt(2)`
D. `4/(5sqrt(2))`

Answer

To find the distance of the plane given by the equation \(3x + 4y - 5z = 60\) from the origin, we can follow these steps:

### Step 1: Rewrite the equation of the plane
The equation of the plane can be rewritten in the standard form:
\[ 3x + 4y - 5z - 60 = 0 \]

### Step 2: Identify the coefficients and the constant
From the equation \(3x + 4y - 5z - 60 = 0\), we can identify:
- Coefficient of \(x\) (A) = 3
- Coefficient of \(y\) (B) = 4
- Coefficient of \(z\) (C) = -5
- Constant term (D) = -60

### Step 3: Use the distance formula from a point to a plane
The formula to calculate the distance \(d\) from a point \(P(x_1, y_1, z_1)\) to the plane \(Ax + By + Cz + D = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

### Step 4: Substitute the coordinates of the origin
For the origin, the coordinates are \(x_1 = 0\), \(y_1 = 0\), and \(z_1 = 0\). Substituting these values into the distance formula:
\[ d = \frac{|3(0) + 4(0) - 5(0) - 60|}{\sqrt{3^2 + 4^2 + (-5)^2}} \]

### Step 5: Simplify the numerator
The numerator simplifies to:
\[ |0 + 0 + 0 - 60| = |-60| = 60 \]

### Step 6: Simplify the denominator
Now, calculate the denominator:
\[ \sqrt{3^2 + 4^2 + (-5)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \]

### Step 7: Calculate the distance
Now substituting back into the distance formula:
\[ d = \frac{60}{\sqrt{50}} = \frac{60}{5\sqrt{2}} = \frac{12}{\sqrt{2}} = 12 \cdot \frac{\sqrt{2}}{2} = 6\sqrt{2} \]

Thus, the distance of the plane from the origin is:
\[ \boxed{6\sqrt{2}} \]