3D Geometry
Planes and Distances
Grade 12
Question:
<p>If a plane p₁ is drawn from the point A(\(\vec{a}\)) and another plane p₂ is drawn from point B(\(\vec{b}\)) parallel to p, then the distance between the planes p₁ and p₂ is:</p>
<p>(a) \(\frac{|(\vec{a} - \vec{b}) \times \vec{n}|}{|\vec{n}|}\)</p>
<p>(b) \(|(\vec{a} - \vec{b}) \times \vec{n}|\)</p>
<p>(c) \(|(\vec{a} - \vec{b}) \cdot \vec{n}|\)</p>
<p>(d) \(\frac{|(\vec{a} - \vec{b}) \cdot \vec{n}|}{|\vec{n}|}\)</p>
Step-by-Step Solution
Key Concept: Distance between parallel planes equals the component of the displacement vector in the direction of the normal, divided by the magnitude of the normal.
Since p_1 and p_2 are parallel to p, they have the same normal vector \(\vec{n}\). The distance between two parallel planes is the projection of the vector joining any point on one plane to any point on the other, onto the normal direction. This distance is \(\frac{|(\vec{a} - \vec{b}) \cdot \vec{n}|}{|\vec{n}|}\).
Correct Answer: d