Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O (0, 0, 0), P ( 3 , − 1 , 0 ) , and Q ( 3 , 1 , 0 ) . Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure). a. Find the coordinates of R . ( Hint: The distance between the centers of any two spheres is 2.) b. Let r ij be the vector from the center of sphere I to the center of sphere J. Find r OP , r OQ , r PQ , r OR , and r PR .
Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O (0, 0, 0), P ( 3 , − 1 , 0 ) , and Q ( 3 , 1 , 0 ) . Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure). a. Find the coordinates of R . ( Hint: The distance between the centers of any two spheres is 2.) b. Let r ij be the vector from the center of sphere I to the center of sphere J. Find r OP , r OQ , r PQ , r OR , and r PR .
Solution Summary: The author explains that the coordinates of R are (2sqrt3,0) and
Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O(0, 0, 0),
P
(
3
,
−
1
,
0
)
, and
Q
(
3
,
1
,
0
)
. Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure).
a. Find the coordinates of R. (Hint: The distance between the centers of any two spheres is 2.)
b. Let rij be the vector from the center of sphere I to the center of sphere J. Find rOP, rOQ, rPQ,rOR, and rPR.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Let v be a vector whose coordinates are given as v = [vx, Vy, Vz. If the
quaternion Q represents a rotation, show that the new, rotated coordinates of v are
given by Q(0, Vx, Vy, Vz)Q*, where (0, vx, Vy, Vz) is a quaternion with zero as its real
component.
A triangle ABC having coordinates A(5,5), B(10,3), C(7,10) is to be scaled two times in x direction and three times in y direction with respect to point A. Find the new coordinates of triangle A’B’C’.
One of the most important mathematical problems through all times has been to find the area of a polygon. For example, real estate areas often had the shape of polygons, and the tax was proportional to the area. Suppose we have some polygon with vertices ("corners") specified by the coordinates (x1,y1), (x2,y2), …, (xn,yn), numbered either in a clockwise or counter-clockwise fashion around the polygon. The area A of the polygon can amazingly be computed by just knowing the boundary coordinates:
A=12|(x1y2+x2y3+⋯+xn−1yn+xny1)−(y1x2+y2x3+⋯+yn−1xn+ynx1)|.(17)
Write a function polygon_area(x, y) that takes two coordinate lists with the vertices as arguments and returns the area.
Test the function on a triangle, a quadrilateral, and a pentagon where you can calculate the area by alternative methods for comparison.
Hint.
Since Python lists and arrays has 0 as their first index, it is wise to rewrite the mathematical formula in terms of vertex coordinates numbered as x0,x1,…,xn−1 and…
Chapter 13 Solutions
Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package
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