Name: ___________________

Math 122 Lab 7: The Curved Cube

Introduction

If you take a cube and spin it rapidly around an axis through opposite vertices, you will see a curved outline appear, as in the picture shown below.

In this lab, you will use vector projections to discover how the cube curves. The idea is that with the cube spinning rapidly, what you see is the combination of points on the cube at their maximum distance from the diagonal. We use vector projections to compute these distances.

What to Turn In

Turn in this worksheet with everything filled in. This will be graded for accuracy and understanding, so good explanations are important. If you need more space, turn in additional pages typed or neatly written. Due Friday at 8:30. Late papers lose 4 points per day late.

Problems

Assume that the cube has side length 1, with , , and that the axis of rotation is through the points (0,0,0) and (1,1,1). For a given point (x,y,z), we want the height h as shown in the picture.

It should be clear that the points on the cube that are farthest from the axis of rotation are on the edges of the cube. So we look at the cube edge by edge. It turns out that the curved part is created by the edge from the point (0,0,1) to the point (0,1,1). Since only the y-coordinate is changing, we can write any point on this edge as (0,y,1) where . Find the component of the vector <0,y,1> along the vector <1,1,1> and write the result here.

Does the component give the value of t or h? Briefly explain your answer.

Note that t, h and <0,y,1> form a right triangle. Use the Pythagorean formula to get an equation involving t, h and y.

Use the component equation found above to substitute for y. This leaves an equation involving t and h. Solve for h.

Graph with . This is the curve! Sketch it here.

Use your component calculation to explain why.

Extra Credit: show that the curve is a piece of a hyperbola.

Repeat the above process to get the h(t) equation for the edge connecting the points (0,0,0) and (0,0,1). (Hint: this part is not curved.)

Repeat the process with the edge connecting (0,0,0) and (1,0,0). Explain why this does not add anything to our picture.

What do you expect will happen with the side from (1,0,0) to (1,1,0)? Show that you are correct.

The third piece of the spinning cube (the straight segment on the right) can be found by repeating our process for the edge connecting (0,1,1) and (1,1,1). This involves some messy algebra, but you end up with (you don’t need to show this!) for . The simplest way to graph this is to find two points and connect them with a line segment. Fill in the blanks and sketch this segment of the curve. Then use information from previous pages to draw the full graph of h(t).

, h = _______ , h = _______