Final answer:
The value of lambda (λ), representing the ratio of the position vector of the orthocentre to the centroid, is 2/3, which corresponds to the answer choice D. The correct answer is λ = 2/3, which corresponds to option D.
Step-by-step explanation:
The student's question pertains to the relationship between the position vector of the orthocentre (denoted as ¯p) and the position vector of the centroid (denoted as ¯g) of a triangle ABC, considering that the circumcenter of the triangle is at the origin. We've been given that ¯p = λg and we need to determine the value of λ corresponding to centroid and orthocentre.
It's important to note that, in a triangle, the centroid is located at 1/3 of the distance from each vertex to the midpoint of the opposite side.
This gives us the relationship that the position vector of the centroid is one-third the sum of the position vectors of the vertices of the triangle when the circumcenter is at the origin.
The position vector of the orthocenter of the triangle ABC, denoted as −p, can be related to the position vector of the centroid, denoted as −g, when the circumcenter is the origin. If −p = λg, we need to find the value of λ. The value of λ can be found by comparing the coordinates of −p and −g. Since the centroid divides each median in a ratio of 2:1, the value of λ is 2.
Thus, the value of λ is equal to the ratio of the lengths of the orthocentre vector to the centroid vector. The correct answer is λ = 2/3, which corresponds to option D.