Question

The length of the perpendicular to the plane \[ \vec r \cdot (i - 2j + 3k) = 14 \] from the origin is

• A. \(\sqrt{7}\) units
• B. \(7\) units
• C. \(14\) units
• D. \(\sqrt{14}\) units

Hint:

For planes passing through origin, distance is zero; otherwise use the standard distance formula.

Answer 1

The Correct Option is D

Step 1: Write the plane in Cartesian form.
Given \[ \vec r \cdot (i - 2j + 3k) = 14 \] This represents the plane \[ x - 2y + 3z - 14 = 0 \]
Step 2: Use the formula for distance from origin to a plane.
The distance of the plane \[ ax + by + cz + d = 0 \] from the origin is \[ \frac{|d|}{\sqrt{a^2 + b^2 + c^2}} \]
Step 3: Substitute the values.
Here \(a=1,\ b=-2,\ c=3,\ d=-14\). \[ \text{Distance} = \frac{14}{\sqrt{1^2 + (-2)^2 + 3^2}} = \frac{14}{\sqrt{14}} = \sqrt{14} \]