Final answer:
The vector components of the diagonal of a rectangle with corners at (2,3) and (16,-1) can be found using the difference in coordinates: Ax = 14 and Ay = -4. The diagonal vector thus can be written as A = 14 i - 4 j.
Step-by-step explanation:
The student's question involves finding the components of a vector that represents the diagonal of a rectangle with the given corners at (2,3) and (16,-1) on a Cartesian coordinate system.
To find the vector components of the diagonal vector, Ax and Ay, we can use the coordinates of the corners. The component Ax is the difference in the x-coordinates, and the component Ay is the difference in the y-coordinates, since the transverse axis is horizontal and the rectangle is auxiliary. Applying this method, Ax = 16 - 2 = 14, and Ay = -1 - 3 = -4. Therefore, the vector A can be represented as A = Ax i + Ay j, or more concretely, A = 14 i - 4 j.
Given the right triangle formed by the components Ax, Ay, and the vector A, we can verify the components using the Pythagorean theorem, ensuring the relationship between the sides of a right triangle. In this context, |A| = √(Ax² + Ay²) which yields the magnitude of the diagonal vector.
Lastly, if the angles were provided, the general form of finding the x and y components of a vector could utilize the equations Ax = A cos θ and Ay = A sin θ where θ is the angle made with the x-axis. However, since no angle is provided in this scenario, we rely on the coordinates directly to determine the components.