3D Geometry
Planes and Distances
Grade 12
Question:
<p>Consider a plane p: \(\vec{r} \cdot \vec{n} = d\) (where \(\vec{n}\) is not a unit vector). There are two points A(\(\vec{a}\)) and B(\(\vec{b}\)) lying on the same side of the plane. If foot of perpendicular from A and B to the plane p are P and Q respectively, then length of PQ is:</p>
<p>(a) \(\frac{|(\vec{b} - \vec{a}) \times \vec{n}|}{|\vec{n}|}\)</p>
<p>(b) \(|(\vec{b} - \vec{a}) \times \vec{n}|\)</p>
<p>(c) \(\frac{|(\vec{b} - \vec{a}) \cdot \vec{n}|}{|\vec{n}|}\)</p>
<p>(d) \(|(\vec{b} - \vec{a}) \cdot \vec{n}|\)</p>
Step-by-Step Solution
Key Concept: The distance PQ equals the component of \(\vec{AB}\) in the direction perpendicular to the plane, which is the projection onto \(\vec{n}\) divided by \(|\vec{n}|\).
The distance between two perpendiculars from A and B to the plane is the projection of \(\vec{AB}\) onto the normal direction \(\vec{n}\). This is given by \(\frac{|(\vec{b} - \vec{a}) \cdot \vec{n}|}{|\vec{n}|}\), which is the formula for distance between parallel lines perpendicular to the plane.
Correct Answer: c