In ∆ABC, OA=3i-5j, OB=-7i+4j and the position vector of centroid G is 2i+j. Find OC.
Make a sketch.
Since position vector like OA = 3i-5j
set A as the point (3,-5), we also have
OB = -7i+4j , placing B at (-7,4)
let D be the midpoint of AB,
D = (-2, -1/2) using the standard way to find midpoint.
Also we know the centroid G is at (2,1)
We also know that the centroid is located 1/3 of the distance of the median measured from D
Let OC be the position vector ai + bj, so that C is (a,b)
vector DG = 1/3 vector DC
<4,3/2> = 1/3 <a+2, b+1/2>
(1/3)(a+2) = 4
a+2=12
a=10
(1/3)(b+1/2) = 3/2
b+1/2 = 9/2
b = 4
so OC = 10i + 4j
check my arithmetic
To find the position vector OC, we need to use the formula for the position vector of the centroid:
OG = 1/3 * (OA + OB + OC)
Given that OG = 2i + j, OA = 3i - 5j, and OB = -7i + 4j, we can substitute these values into the formula:
2i + j = 1/3 * (3i - 5j + (-7i + 4j) + OC)
Next, we can simplify the equation:
2i + j = 1/3 * (-4i - j + OC)
Multiplying both sides of the equation by 3, we get:
6i + 3j = -4i - j + OC
Combining like terms, we have:
10i + 4j = OC
Therefore, the position vector OC is 10i + 4j.
To find OC, we need to determine the position vector of point C.
The centroid G divides each median in a ratio of 2:1. This means that the position vector of G is 2/3 of the distance from the vertex to the opposite midpoint along the median.
Given that the position vector of centroid G is 2i + j, we can set up the following equation to find the position vector of point C:
OG = (2/3)OC
First, let's find the position vector of point O:
OA = 3i - 5j
Now, we can find the position vector of point C:
2/3 * OC = OG
2/3 * OC = 2i + j
To solve for OC, multiply both sides of the equation by 3/2 to isolate OC:
OC = (3/2) * (2i + j)
OC = 3i/2 + 3j/2
Therefore, the position vector of point C, OC, is 3i/2 + 3j/2.