Question

Let \(O\) be the origin and let \(P Q R\) be an arbitrary triangle. The point \(S\) is such that $$ \overrightarrow{O P} \cdot \overrightarrow{O Q}+\overrightarrow{O R} \cdot \overrightarrow{O S}=\overrightarrow{O R} \cdot \overrightarrow{O P}+\overrightarrow{O Q} \cdot \overrightarrow{O S}=\overrightarrow{O Q} \cdot \overrightarrow{O R}+\overrightarrow{O P} \cdot \overrightarrow{O S} $$ Then the triangle \(P Q R\) has \(S\) as its [A] centroid [B] circumcentre [C] incentre [D] orthocenter

Short answer

The point S is the centroid of the triangle PQR since it equidistantly balances the dot products of the vectors originating from O to the vertices of the triangle.

Step-by-step solution

Interpret the vectors and use properties of dot product

The given vectors all originate from the point O. The dot product of two vectors \textbf{A} and \textbf{B}, denoted as \textbf{A} \textbf{\(\textbf{B}\)} \textbf{B}, represents the product of the magnitudes of the vectors and the cosine of the angle between them, \textbf{A} \textbf{\(\textbf{B}\)} \textbf{B} = |\textbf{A}| |\textbf{B}| \textbf{\(\textbf{cos}\)} \textbf{\(\theta\)}, where \textbf{\(\theta\)} is the angle between the vectors. The given information suggests a relationship between angles and sides of the triangle that involves the point S.

Rewrite the equations to reveal relationships

By arranging the equations given, we can see the relationships between the dot products more clearly. Equation 1: \textbf{O}\(\textbf{P}\) \(\textbf{O}\)\textbf{Q}) + (\textbf{O}\(\textbf{R}\) \(\textbf{O}\)\textbf{S}) = (\textbf{O}\(\textbf{R}\) \(\textbf{O}\)\textbf{P}) + (\textbf{O}\(\textbf{Q}\) \(\textbf{O}\)\textbf{S}). Equation 2: (\textbf{O}\(\textbf{R}\) \(\textbf{O}\)\textbf{P}) + (\textbf{O}\(\textbf{Q}\) \(\textbf{O}\)\textbf{S}) = (\textbf{O}\(\textbf{Q}\) \(\textbf{O}\)\textbf{R}) + (\textbf{O}\(\textbf{P}\) \(\textbf{O}\)\textbf{S}). Equation 3: (\textbf{O}\(\textbf{Q}\) \(\textbf{O}\)\textbf{R}) + (\textbf{O}\(\textbf{P}\) \(\textbf{O}\)\textbf{S}) = (\textbf{O}\(\textbf{P}\) \(\textbf{O}\)\textbf{Q}) + (\textbf{O}\(\textbf{R}\) \(\textbf{O}\)\textbf{S}). In each equation, the dot products suggest that S is equidistant from the sides of the triangle, meaning all angles to the sides are equal, which is a property observed in the centroid of a triangle.

Identify the property of point S

The centroid of a triangle is the point where the three medians intersect, and is also the balance point of the triangle. It is equidistant from all the vertices. Comparing this to the relationships derived from the dot products, we can see that S must be the point that satisfies these conditions -- equidistant from all sides and balancing the dot products equally. Thus, S is the centroid of triangle PQR.

Choose the correct option

Based on the properties and relationship between the point S and the vertices of triangle PQR derived from the dot products, we conclude that S is the centroid of triangle PQR. Hence, the correct answer is [A] centroid.

Key concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra that reveals the relationship between two vectors. It's defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. The formula is expressed as \( \textbf{A} \cdot \textbf{B} = |\textbf{A}| |\textbf{B}| \cos(\theta) \) where \( \textbf{A} \) and \( \textbf{B} \) are vectors and \( \theta \) is the angle between them. When dealing with vectors originating from the same point in a geometric context, like in the case of triangle vertices emerging from the origin, the dot product can give us insights about the angles and lengths of sides in relation to each other.

When applied to the problem at hand, the dot product reveals certain symmetries that are crucial in identifying the unique position of point \( S \) in relation to triangle \( PQR \) which helps us deduce its geometrical significance.

Centroid of a Triangle

The centroid of a triangle is where the medians intersect, effectively becoming the triangle's center of gravity. Each median of a triangle is the line drawn from a vertex to the midpoint of the opposite side. The centroid, often denoted by \( G \) is remarkable for dividing each median into segments in a 2:1 ratio, where the longer part is closest to the vertex. This attribute signifies that the centroid is the balancing point for the triangle's surface.

In the context of the exercise, the centroid is conceptually linked to the notion of equilibrium among the vectors originating from the origin, hence equalizing the dot products. This understanding aligns with the given vector equations which suggest that point \( S \) represents the centroid by balancing out the scalar quantities derived from the dot products.

Medians of a Triangle

Medians in a triangle are lines from each vertex to the midpoint of the opposite side. Each triangle has exactly three medians, and these lines are of special interest since they always intersect at a single point, the centroid, regardless of the triangle’s type. A median essentially bisects the area of a triangle, contributing to the unique role of the centroid in being the 'center of mass'.

This geometric property can be visualized in the context of vectors emanating from a common origin point, as they represent the sides of the triangle stretching out to their respective vertices. It follows that the point of concurrency of the medians, or the centroid, determines a special balance in the context of vectors and helps to underscore the equalizing nature of point \( S \) in our problem, indicating why the medians are central to identifying the triangle's centroid.