The centroid of the triangle OAB is denoted by G. If O is the origin and OA=4i + 3j ,OB=6i-j. Find OG in terms of the unit vectors, I and j.

See:

http://www.jiskha.com/display.cgi?id=1416936762

To find the centroid G, we need to find the average of the coordinates of points O, A, and B.

Given that O is the origin (0, 0), OA = 4i + 3j, and OB = 6i - j, we can find the coordinates of point A and B.

Coordinates of point A:
Since vector OA represents the displacement from O to A, the coordinates of point A can be obtained by adding the displacement vector OA to the coordinates of point O.

OA = 4i + 3j
Coordinates of A = Coordinates of O + OA = (0, 0) + (4, 3) = (4, 3)

Coordinates of point B:
Similarly, we can find the coordinates of point B by adding the displacement vector OB to the coordinates of point O.

OB = 6i - j
Coordinates of B = Coordinates of O + OB = (0, 0) + (6, -1) = (6, -1)

Now that we have the coordinates of points O, A, and B, we can find the coordinates of the centroid G.

Coordinates of the centroid G:
The coordinates of the centroid G can be found by taking the average of the x-coordinates and the average of the y-coordinates of points O, A, and B.

Average x-coordinate = (x-coordinate of O + x-coordinate of A + x-coordinate of B) / 3
= (0 + 4 + 6) / 3
= 10 / 3

Average y-coordinate = (y-coordinate of O + y-coordinate of A + y-coordinate of B) / 3
= (0 + 3 - 1) / 3
= 2 / 3

Therefore, the coordinates of the centroid G are (10/3, 2/3).

Finally, to find OG in terms of the unit vectors i and j, we can calculate the displacement vector between O and G:

OG = (x-coordinate of G - x-coordinate of O) * i + (y-coordinate of G - y-coordinate of O) * j
= (10/3 - 0) * i + (2/3 - 0) * j
= (10/3) * i + (2/3) * j

Thus, OG is equal to (10/3) * i + (2/3) * j.