MAT188-HW1 GRADESCOPE TEMPLATE 1, Sep 21, 11:59 PM Page 4 of 12 (5) Let ẞ denote the angle between OE and OF. Intuitively, what is the relationship between a and B? Investigate your guess mathematically. (6) Use dot product to verify that AE is perpendicular to AB. MAT188-HW1 GRADESCOPE TEMPLATE 1, Sep 21, 11:59 PM 5 (8) There are infinitely many planes in R³ that are perpendicular to l₁. Find the equation of two such planes: P₁ that passes through A, and P2 that passes through I. Describe the planes in set notation. How do these two planes compare? How do their equations compare? Explain. P₁ = { P2 = { (9) Suppose you are at the origin. Describe how you can get to the tip of M = (1/2, 0, 1/2) on P1, using OA and OB. (7) Consider the line l₁ that goes through A and E. Find the vector form of this line and describe it in proper set notation. (10) Suppose M' is the point opposite to M on the flip side of the red piece, on the plane P2. Starting from the origin, how can you get to M' using OA, OB and v. l₁ = { MAT188-HW1 GRADESCOPE TEMPLATE 1, Sep 21, 11:59 PM 2 Problem 1. The Japanese puzzle designer Yasuhiro Hashimoto has a puzzle made from three identical pieces which are to be put together to form a 3 × 3 × 3 cube. The pieces are easy to design, however fitting them in a cube is surprisingly difficult. Here is what the pieces look like when placed on a MAT188 voting booklet: 3 MAT188-HW1 GRADESCOPE TEMPLATE 1, Sep 21, 11:59 PM (2) From the angle showed in the figure above, the points A, B, C, and O are corners of the front side of the piece (which looks like the letter C). There is only one corner labeled on the back side of this letter C shape, that is I. Let C denote the set of all vectors (whose end point) draws the front of the letter C in the figure. Describe this set using set notation. You may need to create a number of sets, each describing a portion of the image, and then take their union. tte Set up a coordinate system so that a corner of the red piece sits at the origin as showed in the image. C = (3) Find a vector that when added to the position vector of any point on the front side of the letter C gives the corresponding point on the back side. For instance, v added to the position vector of B should yield that of I. y (1) Find the position vectors of the points A through I. You just need to write the vectors using the correct notation. No explanation needed. √ = (4) Let a denote the angle between OB and OI. Use the dot product to find a. OA = OB = OC = OD= OE= OF OG = OH = σI =