The points A(1, -2, 4), B(3, 5, 7), C(4, 6, 8) are three of four vertices of parallelogram ABCD. Determine the coordinates of vertex D
Could someone please tell me to go about solving this question. Thanks a lot.
To find the coordinates of vertex D, we can use the fact that opposite sides of a parallelogram are parallel and equal in length.
Step 1: Determine the vector represented by the line segment AB.
- Subtract the coordinates of point A from the coordinates of point B to find the change in x, y, and z values.
- The vector AB is represented as: vector AB = (x2 - x1, y2 - y1, z2 - z1).
Step 2: Add the vector AB to the coordinates of point C to determine the coordinates of vertex D.
- Add the x, y, and z values of the vector AB to the x, y, and z values of point C.
- The coordinates of vertex D are represented by: (x1 + x-component of vector AB, y1 + y-component of vector AB, z1 + z-component of vector AB).
Let's apply these steps to the given information.
Step 1: Find vector AB.
- AB = (3 - 1, 5 - (-2), 7 - 4)
= (2, 7, 3)
Step 2: Add vector AB to the coordinates of point C.
- D = (4 + 2, 6 + 7, 8 + 3)
= (6, 13, 11)
Therefore, the coordinates of vertex D are (6, 13, 11).