In ∆ABC, AD median of BC where position vectors of A and D are 3i-2j and -3i-4j respectively. Find the position vector of centroid G of the ∆ABC.
Well, well, well, it seems like we have a geometry question here. Let me calculate it for you and try to put a smile on your face while doing so.
To find the position vector of the centroid G of triangle ABC, we can use the formula (A + B + C)/3.
First, let's find the position vector of point B. Since D is the midpoint of BC, we can use the midpoint formula: B = (A + D)/2.
Let's do some math magic:
B = (3i - 2j + (-3i - 4j))/2
= (0i - 6j)/2
= 0i - 3j
Now, let's find the position vector of point G, the centroid:
G = (A + B + C)/3
But hold on a second, we don't have the position vector for point C. Well, luckily for us, the centroid divides the median in a 2:1 ratio. So, if D is the midpoint of BC, we can say that C is 2D.
Here's G... wait for it... wait for it...
G = (3i - 2j + 0i - 3j + 2(-3i - 4j))/3
= (3i - 2j - 6i - 8j)/3
= (-3i - 10j)/3
= -i - (10/3)j
And there you have it, the position vector of the centroid G is -i - (10/3)j. Ta-da!
The centroid is located 2/3 of the length of a median, measured from the vertex
OA = (3, -2)
OD = (-3, -4)
we know AG : GD = 2 : 1
OG =(1/3)(3,-2) + (2/3)(-3,-4)
= (1, -2/3) + (-2, -8,3)
= (-1, -10/3)
or -i - (10/3)j
To find the position vector of centroid G of ∆ABC, we need to determine the position vector of point B.
Since AD is a median, point D divides side BC into two equal parts: BD and DC.
Let's find the position vector of point B using the position vectors of A and D.
Position vector of B = 2 * Position vector of D - Position vector of A
We have:
Position vector of B = 2 * (-3i - 4j) - (3i - 2j)
= -6i - 8j - 3i + 2j
= -9i - 6j
Now, let's find the position vector of the centroid G.
Position vector of G = (Position vector of A + Position vector of B + Position vector of C) / 3
We know that the centroid G divides the medians (AD, BE, and CF) in a 2:1 ratio.
Since AD is a median, the position vector of G will be:
Position vector of G = (Position vector of A + Position vector of D) / 3
Substituting the values:
Position vector of G = (3i - 2j + -3i - 4j) / 3
= 0i - 6j / 3
= 0i - 2j
= -2j
Therefore, the position vector of the centroid G of ∆ABC is -2j.
To find the position vector of the centroid G of triangle ABC, we can use the concept that the centroid of a triangle is the average of the positions vectors of its three vertices.
Let's denote the position vectors of points A, B, and C as vector A, vector B, and vector C, respectively.
Given:
Position vector of point A, vector A = 3i - 2j
Position vector of point D, vector D = -3i - 4j
Since AD is a median of ∆ABC, we know that point D divides segment BC into two equal parts. Therefore, the position vector of point B, vector B, will be the negative of the position vector of point C, vector C.
Now, using this information, we can proceed to find the position vector of point C.
1. Find the position vector of point B, vector B:
- Since point D divides BC into two equal parts, we can find the position vector of point B as the average of vectors D and C.
- Vector B = (Vector D + Vector C) / 2
- Since Vector C = -Vector B, we can write the equation as:
Vector B = (Vector D - Vector B) / 2
- Simplifying further, we can rearrange the equation to solve for Vector B:
2 * Vector B = Vector D - Vector B
- Adding Vector B to both sides:
3 * Vector B = Vector D
- Dividing both sides by 3:
Vector B = Vector D / 3
- Substituting the given value of Vector D:
Vector B = (-3i - 4j) / 3
Vector B = -i - (4/3)j
2. Find the position vector of point C:
- Since Vector C = -Vector B, we can find the position vector of point C as the negative of Vector B:
Vector C = -Vector B
- Substituting the value of Vector B obtained in the previous step:
Vector C = -(-i - (4/3)j)
Vector C = i + (4/3)j
3. Find the position vector of the centroid G:
- The centroid G is the average of the position vectors of points A, B, and C.
Vector G = (Vector A + Vector B + Vector C) / 3
- Substituting the given values of Vector A, Vector B, and Vector C:
Vector G = (3i - 2j + (-i - (4/3)j) + (i + (4/3)j)) / 3
Vector G = (3i - 2j - i - (4/3)j + i + (4/3)j) / 3
- Simplifying the equation:
Vector G = (3i - i + i) / 3 + (-2j - (4/3)j + (4/3)j) / 3
Vector G = (3i) / 3 + (-(2/3)j) / 3
- Simplifying further:
Vector G = i - (2/3)j
Therefore, the position vector of the centroid G of triangle ABC is i - (2/3)j.