3D Geometry
Reflection of Points
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. Reflection of A(\(\vec{a}\)) in the plane p has the position vector:</p>
<p>(a) \(\vec{a} + \frac{2}{|\vec{n}|^2}(d - \vec{a} \cdot \vec{n}) \vec{n}\)</p>
<p>(b) \(\vec{a} - \frac{1}{|\vec{n}|^2}(d - \vec{a} \cdot \vec{n}) \vec{n}\)</p>
<p>(c) \(\vec{a} + \frac{2}{|\vec{n}|^2}(d + \vec{a} \cdot \vec{n}) \vec{n}\)</p>
<p>(d) \(\vec{a} + \frac{2}{|\vec{n}|^2} \vec{n}\)</p>
Step-by-Step Solution
Key Concept: Reflection of a point in a plane involves finding the foot of perpendicular and then extending twice that distance on the opposite side.
To find the reflection of A in the plane, we first find the perpendicular distance from A to the plane: \(dist = \frac{d - \vec{a} \cdot \vec{n}}{|\vec{n}|}\). The reflection is obtained by moving A twice this distance in the direction of \(\vec{n}\). Therefore, the position vector of reflection is \(\vec{a} + \frac{2}{|\vec{n}|^2}(d - \vec{a} \cdot \vec{n}) \vec{n}\).
Correct Answer: a