Question

The perpendicular distance of the plane \( r \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) = 78 \) from the origin is __________.

• A. \( 4 \)
• B. \( 5 \)
• C. \( 6 \)
• D. \( 8 \)

Hint:

The perpendicular distance of a plane \( Ax + By + Cz = D \) from the origin is given by: \[ d = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}} \]

Answer 1

The Correct Option is C

Step 1: Formula for Perpendicular Distance
The perpendicular distance \( d \) from the origin to the plane \( Ax + By + Cz = D \) is calculated using the formula: \[ d = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}} \] Step 2: Substituting the Given Values
Given that: \[ A = 3, \quad B = 4, \quad C = 12, \quad D = 78 \] we substitute these values into the formula: \[ d = \frac{|78|}{\sqrt{3^2 + 4^2 + 12^2}} \] \[ = \frac{78}{\sqrt{169}} = \frac{78}{13} = 6 \]