Misconceptions or Tabula Rasa. Can’t think of too many misconceptions here and would expect students to have a reasonable understanding of the point concept and to understand what the coordinates of a point are. The main purpose of this material is to connect to something the students are familiar with and establish common ground from which we can depart together into less familiar waters.

• Begin. Still to be done. A video or cartoon that illustrates the importance of position, the idea of a target. Could be illustrated with bow and arrow shooting and landing a spacecraft on the moon. Tool Tip: "The role of position in trajectories and targets".
• Explain It. Still be done. Have a simulation that lets you draw a point, move the point, draw a pair of axes, move and rotate the axes, and that will display the coordinates of the point in the form (…, …). It will be easy for students to accept that we require two coordinates to represent in point in 2-d, three in 3-d. (Position vectors are not introduced at this time.) It will be explained that a point can be thought of two ways: geometrically (graphically via a dot) and analytically (via numbers). The geometrical concept of a point is something absolute, the analytical description is relative to the axes chosen. The simulation will illustrate this. The 1-d case is also introduced. Here, we have only one axis, whose direction and orgin can be chosen arbitrarily. Tool Tip: "Points and their coordinates"
• Simulate It. Still to be done. The Simulation from Explain It by itself. Tool Tip: "Points and changeable coordinate axes".
• Test Yourself. Still to be done. Question on analytic description of a point and its dependence on the axes vs. the geometric invariant property of a point. Tool Tip: "Points are absolute and coordinates are relative."
• Get Information. Still to be done. Summary of the main ideas from Explain It without the simulation, but maybe a graph. Tool Tip: "Points and their coordinates".

Misconceptions or Tabula Rasa. The latter in this case. Position vector is a new concept that needs to be introduced. A misconception that can arise is that position vectors are like other vectors and can be added or scalar multiplied. It should be made clear that that is not possible so that position vectors are not really vectors, only look like vectors and for that reason share the name. They are displacement vectors attached to a given point.

Should we very briefly mention the notion of "position space" as opposed to other spaces, e.g., "velocity space"? Position space has two kinds of elements: points and displacement vectors. Mathematically, such a space is called an affine space. Its structure is distinct from that of a vector space, for which velocity space is an example. These spaces can be easily distinguished from each other because their elements have different physical dimension.

The material under Position Vector requires Displacements as a prerequisite.

• Explain It. Still to be done. Use simulation from Position/Basic Facts/Explain It, but modified so that students can draw vectors in it as well. Have students draw a displacement vector from the origin of a pair of axes to a given point. Call this the position vector of that point and show that its scalar components are identical with the coordinates of the point. Explain the notion of "trajectory", with simulation showing position vectors to points on the trajectory clicked by student.

Have student click two points on the trajectory: system will show the two position vectors and the corresponding displacement vector. Adding the displacement vector to the first position vector gives the second position vector. One obtains the displacement vector as the change (difference) of the two position vectors. Generalize to difference of vectors. Use components to work out the difference. Refer to Vectors and tail-to-tail construction of difference.

Tool Tip: "Points, position vectors, trajectories, changes in position vectors".

• Simulate It. Still to be done. Use simulation from Explain It. Tool Tip: "Points and position vectors".
• Test Yourself. Still to be done. A couple of questions on notation (components of position vector are denoted x and y), position vectors and displacements along trajectories. Displacement vectors as difference of position vectors. Tool Tip: "Position vectors, notation, trajectories".
• Get Information. Still to be done. The content of Explain It without simulation. Tool Tip: "Points, position vectors, trajectories".