63. The Parallelogram Law states that |a + b ² + | a - b ² = 2 | a² + 2|b|² (a) Give a geometric interpretation of the Parallelogram Law. (b) Prove the Parallelogram Law. (See the hint in Exercise 62.)
Displacement, Velocity and Acceleration
In classical mechanics, kinematics deals with the motion of a particle. It deals only with the position, velocity, acceleration, and displacement of a particle. It has no concern about the source of motion.
Linear Displacement
The term "displacement" refers to when something shifts away from its original "location," and "linear" refers to a straight line. As a result, “Linear Displacement” can be described as the movement of an object in a straight line along a single axis, for example, from side to side or up and down. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Linear displacement is usually measured in millimeters or inches and may be positive or negative.
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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).]
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H
H
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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%2F03003359-f942-4611-9e89-2fa0f163f442%2Feivy295_processed.jpeg&w=3840&q=75)
63)
Given that,
the parallelogram law of the vectors and is
a) To draw, the vector representation of the parallelogram theorem.
b) To prove the parallelogram theorem.
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