How Does Position Vector Calculation Work in Geometry?

[Procrastinate]

AD = AB + 2/3BC
OD - OA = OB - OA + 2/3(OC - OB)
OD = 1/3OB + 2/3OC

BE = BC + 3/4CA
BE = OC - OB + 3/4 OA - 3/4OC
OE = 1/4OC - OB + 3/4OA

Could someone please tell me if I am on the right track with this?
I am also stuck on the finding the position vector G. Whenever I attempt to find it, the G always cancels out leaving me with no equation left.

 

[Dick]

Your doing ok so far. So if you represent each position vector as they suggested, e.g. OA=a, OB=b, OC=c, etc, you've got e=(3/4)a+(1/4)c and d=(1/3)b+(2/3)c, right? The line between A and D is given by a+t*(d-a) where t is a parameter which describes where you are on the line (e.g. if t=0 you are at a and if t=1 you are at d). Similarly the line between B and E is given by b+s*(e-d) for a different parameter s. You want to find the intersection of those two lines. I.e. find values of t and s that give the same point.

 

[Architeuthis Dux]

I cannot determine if you are on the right track without more context or information. However, I can provide some general feedback on the equations you have provided.

Firstly, the equations given appear to be position vector equations, which describe the location of a point in space relative to a reference point (usually the origin). These equations are typically used in geometry and physics to describe the position of objects in a coordinate system.

In the first equation, AD = AB + 2/3BC, it is important to note that position vectors are typically written with an arrow on top (e.g. →AD) to indicate that it is a vector quantity. The equation is saying that the position vector for point D is equal to the position vector for point A plus 2/3 of the position vector for point B. This means that point D is located 2/3 of the distance between points A and B.

The second equation, OD - OA = OB - OA + 2/3(OC - OB), is showing that the position vector for point D can also be found by subtracting the position vector for point A from the sum of the position vectors for points B and C. This is a useful property of position vectors, as it allows us to find the position of a point relative to other points in space.

Moving on to the third equation, BE = BC + 3/4CA, it follows the same pattern as the first equation, stating that the position vector for point E is equal to the position vector for point C plus 3/4 of the position vector for point A. Again, this means that point E is located 3/4 of the distance between points C and A.

The last equation, OE = 1/4OC - OB + 3/4OA, is similar to the second equation, showing that the position vector for point E can also be found by subtracting the position vector for point B from the sum of 1/4 of the position vector for point C and 3/4 of the position vector for point A.

As for finding the position vector for point G, it is difficult to provide guidance without more information about the problem you are trying to solve. It could be that the position vector for point G cannot be determined using the given information, or there may be a mistake in the equations provided. It may be helpful to review the problem and any given information to see if there are