Final answer:
To find the endpoints A and B from the midpoint (2,1) and vector V = 2i + 3j, a system of equations based on the properties of midpoints and vectors is set up and solved, resulting in A being (1, -0.5) and B being (3, 2.5).
Step-by-step explanation:
To find the points A and B given the midpoint (2,1) and vector V representing segment AB as V = 2i + 3j, we first note that the midpoint of a segment in two dimensions is given by the average of the x-coordinates and y-coordinates of the endpoints. So, if A is (x1, y1) and B is (x2, y2), the midpoint M would be ((x1 + x2)/2, (y1 + y2)/2). Given that M is (2,1), we can then set up a system of equations:
- (x1 + x2)/2 = 2
- (y1 + y2)/2 = 1
aaAdditionally, since vector V represents the segment from A to B, we can express it as:
- V = (x2 - x1)i + (y2 - y1)j
Given V as 2i + 3j, this means x2 - x1 = 2 and y2 - y1 = 3.
Using these equations, we can solve for the coordinates of A and B.
Finding Point A
Let's assume point A to be (2 - a, 1 - b) because we know that point B must be (2 + a, 1 + b) to satisfy the midpoint condition. Using the equations derived from the vector V, we find that:
- a = 2 / 2 => a = 1
- b = 3 / 2 => b = 1.5
Therefore, point A is (1, -0.5).
Finding Point B
Given A, we can immediately determine that B is (3, 2.5) because B is just a translation of the vector (2i + 3j) from A.