57. A molecule of methane, CH4, is structured with the four hydro- gen atoms at the vertices of a regular tetrahedron and the car- bon atom at the centroid. The bond angle is the angle formed by the H-C-H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about 109.5°. [Hint: Take the vertices of the tetrahedron to be the points (1, 0, 0), (0, 1, 0), (0, 0, 1), and (1, 1, 1), as shown in the figure. Then the centroid is (2)] XA H H H H
Kinetic Theory of Gas
The Kinetic Theory of gases is a classical model of gases, according to which gases are composed of molecules/particles that are in random motion. While undergoing this random motion, kinetic energy in molecules can assume random velocity across all directions. It also says that the constituent particles/molecules undergo elastic collision, which means that the total kinetic energy remains constant before and after the collision. The average kinetic energy of the particles also determines the pressure of the gas.
P-V Diagram
A P-V diagram is a very important tool of the branch of physics known as thermodynamics, which is used to analyze the working and hence the efficiency of thermodynamic engines. As the name suggests, it is used to measure the changes in pressure (P) and volume (V) corresponding to the thermodynamic system under study. The P-V diagram is used as an indicator diagram to control the given thermodynamic system.
57
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CHAPTER 12 Vectors and the Geometry of Space
56. Find the angle between a diagonal of a cube and a diagonal of
one of its faces.
57. A molecule of methane, CH4, is structured with the four hydro-
gen atoms at the vertices of a regular tetrahedron and the car-
bon atom at the centroid. The bond angle is the angle formed
by the H-C-H combination; it is the angle between the lines
that join the carbon atom to two of the hydrogen atoms. Show
that the bond angle is about 109.5°. [Hint: Take the vertices of
the tetrahedron to be the points (1, 0, 0), (0, 1, 0), (0, 0, 1), and
(1, 1, 1), as shown in the figure. Then the centroid is (2,2,2).]
XX
H
ZA
H
H
H
y
58. If c = |a|b+ |b|a, where a, b, and c are all nonzero vectors,
show that c bisects the angle between a and b.
59. Prove Properties 2, 4, and 5 of the dot product (Theorem 2).
60. Suppose that all sides of a quadrilateral are equal in length and
opposite sides are parallel. Use vector methods to show that the
diagonals are perpendicular.
61. Use Theorem 3 to prove the Cauchy-Schwarz Inequality:
a b ≤|a|b|
.
62. The Triangle Inequality for vectors is
a + b ≤|a| + | b |
(a) Give a geometric interpretation of the Triangle Inequality.
(b) Use the Cauchy-Schwarz Inequality from Exercise 61 to
the Triangle Inequality. [Hint: Use the fact that
prove
a + b ² = (a + b) · (a + b) and use Property 3 of the
dot product.]
63. The Parallelogram Law states that
| a + b |² + | a − b ² = 2|a|² + 2|b|²
(a) Give a geometric interpretation of the Parallelogram Law.
(b) Prove the Parallelogram Law. (See the hint in Exercise 62.)
64. Show that if u + vand u- v are orthogonal, then the vectors
u and v must have the same length.
65. If 0 is the angle between vectors a and b, show that
projab
.
projь a = (a - b) cos²0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90858446-1c51-4445-91b0-44f008564063%2Ff2febff0-d40d-4876-857c-f1017e65c18c%2Fznmt7ys_processed.jpeg&w=3840&q=75)
We are assuming that the four vertices of the tetrahedron are (1,0,0) ; (0,1,0) ; (0,0,1) and (1,1,1)
and the centroid of the tetrahedron is
Let, v be the vector from the centroid to the point (0,1,0)
and w be the vector from the centroid to the point (0,0,1).
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