if abc oa = 4i - 5j , ob = 6i - 4j and the position vector of centroid G is og = 2i j find oc
not sure what "2i j" means ...
To find the position vector OC, we need to use the fact that the centroid divides the line segment joining points A and B in a 2:1 ratio.
The formula for finding the position vector of a point dividing a line segment in a given ratio is:
PC = (m * PB + n * PA) / (m + n)
Where PC is the position vector of the point C, PB is the position vector of point B, PA is the position vector of point A, and m and n are the ratio numbers.
In this case, we want to find the position vector of point C (OC) on the line segment joining points A (OA) and B (OB), with the ratio of 2:1. Let's calculate it step by step:
Step 1: Calculate the position vector of point A (OA):
OA = 4i - 5j
Step 2: Calculate the position vector of point B (OB):
OB = 6i - 4j
Step 3: Calculate the position vector of point C (OC) using the formula mentioned above:
OC = (2 * OB + 1 * OA) / (2 + 1)
= (2 * (6i - 4j) + 1 * (4i - 5j)) / 3
= (12i - 8j + 4i - 5j) / 3
= (16i - 13j) / 3
Therefore, the position vector OC is (16i - 13j) / 3.
To find the position vector OC, we need to find the position vector OG and subtract it from the position vector OG.
Given:
ABC is a triangle with position vectors:
OA = 4i - 5j
OB = 6i - 4j
The position vector of the centroid G is OG = 2i + j.
To find OC, we can use the fact that the centroid divides the median in a 2:1 ratio.
The position vector to the centroid G (OG) is two-thirds of the position vector to C (OC).
Step 1: Find OC
OC = OG * (2/3)
Step 2: Calculate OG * (2/3):
OG * (2/3) = (2/3)*(2i + j)
Step 3: Distribute the scalar (2/3) to the vector components:
OG * (2/3) = (2/3)*(2i) + (2/3)*(j)
OG * (2/3) = (4/3)i + (2/3)j
Step 4: Substitute the result back into the equation for OC:
OC = (4/3)i + (2/3)j
Therefore, the position vector OC is (4/3)i + (2/3)j.