Answer :
To map Figure D onto Figure E, perform two translations to align the centers of mass and then apply a scaling transformation to resize Figure D.
To map Figure D onto Figure E, you can use a sequence of geometric transformations, such as translations, rotations, and reflections. Let's break it down into steps:
Translation:
First, perform a translation to move Figure D to the right and up to align its center of mass with Figure E. Find the translation vector by calculating the difference in x and y coordinates of the center of mass of both figures. The center of mass for Figure D is ((-3 - 4 - 3 - 2 - 8) / 5, (2 + 4 + 6 + 7 + 7) / 5) = (-4, 5). The center of mass for Figure E is ((7 + 9 + 11 + 12) / 4, (3 + 4 + 3 + 2) / 4) = (9.75, 3).
Translation vector = (9.75 - (-4), 3 - 5) = (13.75, -2)
Translation:
Apply the calculated translation vector to each point in Figure D to move it to its new position.
New coordinates for Figure D:
(-3 + 13.75, 2 - 2) = (10.75, 0)
(-4 + 13.75, 4 - 2) = (9.75, 2)
(-3 + 13.75, 6 - 2) = (10.75, 4)
(-2 + 13.75, 7 - 2) = (11.75, 5)
(-8 + 13.75, 7 - 2) = (5.75, 5)
Scaling:
Finally, perform a uniform scaling to resize Figure D to match the size of Figure E. You can calculate the scaling factor by comparing the lengths of corresponding sides. For example, the distance between the first and second points in Figure D is sqrt((9.75 - 10.75)^2 + (2 - 0)^2) = 1, while in Figure E, it's √((9 - 7)^2 + (4 - 3)^2) = √(5). The scaling factor is sqrt(5).
Apply the scaling factor to each point in Figure D:
New coordinates for Figure D:
(10.75 * √(5), 0 * √(5)) = (15.13, 0)
(9.75 * √(5), 2 * √(5)) = (13.43, 6.46)
(10.75 * √(5), 4 *√(5)) = (15.13, 12.92)
(11.75 * √(5), 5 * √(5)) = (16.83, 16.15)
(5.75 * √(5), 5 * √(5)) = (8.66, 16.15)
After these transformations (two translations and a scaling), Figure D will be mapped onto Figure E.
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Answer:
According to the diagram, is the polar angle (the "vertical" angle made with the positive z-axis) and is the azimuthal angle (the "horizontal" angle made with the positive x-axis), so the convention used here is to take
Then for the spherical point (1, π/4, π/2), we have the corresponding Cartesian point (x, y, z), where
Step-by-step explanation:
