1. The vectors AB and OD are equal vectors. Click and drag the points A and B to verify that this appears to be true.
2. Determine the distances between points A and C and between points B and C ( Use the checkbox "Show lengths"). These distances are the magnitudes of the horizontal and vertical components respectively of vector AB.
3. Select the point D and measure its coordinates (Use the checkbox "Show Points"). Compare the coordinates of D to the magnitudes of AC and CB.
4. How do the directions of the vectors AC and CB compare to the signs of the coordinates of point D ?
5. How do the horizontal and vertical components of the translated vector OD compare to the coordinates of point D? Comment on direction as well as magnitude. The vector OD is referred to as a position vector, and can also be referred to as the vector d.
6. Given the vector AB with points A(xA, yA) and B(xB, yB), suggest a formula to determine the coordinates of the point D so that the vector OD is equal in length and direction to AB. Test your conjectures by changing around the vector AB so that the position vector points to locations in all four quadrants.
7. A vector that has been translated so that its tail is at the origin of the Cartesian plane is called a position vector. Position vectors have interesting characteristics that make them convenient to work with. For example, given the coordinates of any position vector, how would you find the magnitude of that vector?
8. Summarize the results of your investigation of these vectors.