41. The volume of a parallelepiped is equal to the product of its altitude and the area of its base. Take the

base to be the parallelogram formed by the vectors 
b and 
c. Its area is bc sin φ, where φ is the angle
between 
b and 
c. This is just the magnitude of the vector (cross)product 
b

× 
c. The altitude of the

parallelepiped is a cos θ, where θ is the angle between 
a and the normal to the plane of 
b and 
c. Since

b ×
c is normal to that plane, θ is the angle between 
a and 
b ×
c. Thus, the volume of the parallelepiped

is V = a

|
b × 
c| cos θ = 
a · (
b × 
c).

Document Outline

• Main Menu
• Chapter 1 Measurement
• Chapter 2 Motion Along a Straight Line
• Chapter 3 Vectors
  • 3.1 - 3.10
    • 3.1
    • 3.2
    • 3.3
    • 3.4
    • 3.5
    • 3.6
    • 3.7
    • 3.8
    • 3.9
    • 3.10
  • 3.11 - 3.20
    • 3.11
    • 3.12
    • 3.13
    • 3.14
    • 3.15
    • 3.16
    • 3.17
    • 3.18
    • 3.19
    • 3.20
  • 3.21 - 3.30
    • 3.21
    • 3.22
    • 3.23
    • 3.24
    • 3.25
    • 3.26
    • 3.27
    • 3.28
    • 3.29
    • 3.30
  • 3.31 - 3.40
    • 3.31
    • 3.32
    • 3.33
    • 3.34
    • 3.35
    • 3.36
    • 3.37
    • 3.38
    • 3.39
    • 3.40
  • 3.41 - 3.50
    • 3.41
    • 3.42
    • 3.43
    • 3.44
    • 3.45
    • 3.46
    • 3.47
    • 3.48
    • 3.49
    • 3.50
  • 3.51 - 3.60
    • 3.51
    • 3.52
    • 3.53
    • 3.54
    • 3.55
    • 3.56
    • 3.57
    • 3.58
    • 3.59
    • 3.60
  • 3.61 - 3.62
    • 3.61
    • 3.62
• Chapter 4 Motion in Two and Three Dimensions
• Chapter 5 Force and Motion I
• Chapter 6 Force and Motion II
• Chapter 7 Kinetic Energy and Work
• Chapter 8 Potential Energy and Conservation of Energy
• Chapter 9 Systems of Particles
• Chapter 10 Collisions
• Chapter 11 Rotation
• Chapter 12 Rolling, Torque, and Angular Momentum
• Chapter 13 Equilibrium and Elasticity
• Chapter 14 Gravitation
• Chapter 15 Fluids
• Chapter 16 Oscillations
• Chapter 17 Waves—I
• Chapter 18 Waves—II
• Chapter 19 Temperature, Heat, and the First Law of Thermodynamics
• Chapter 20 The Kinetic Theory of Gases
• Chapter 21 Entropy and the Second Law of Thermodynamics
• Chapter 22 Electric Charge
• Chapter 23 Electric Fields
• Chapter 24 Gauss’ Law
• Chapter 25 Electric Potential
• Chapter 26 Capacitance
• Chapter 27 Current and Resistance
• Chapter 28 Circuits
• Chapter 29 Magnetic Fields
• Chapter 30 Magnetic Fields Due to Currents
• Chapter 31 Induction and Inductance
• Chapter 32 Magnetism of Matter: Maxwell’s Equation
• Chapter 33 Electromagnetic Oscillations and Alternating Current
• Chapter 34 Electromagnetic Waves
• Chapter 35 Images
• Chapter 36 Interference
• Chapter 37 Diffraction
• Chapter 38 Special Theory of Relativity
• Chapter 39 Photons and Matter Waves
• Chapter 40 More About Matter Waves
• Chapter 41 All About AtomsChapter 42 Conduction of Electricity in Solids
• Chapter 42 Conduction of Electricity in Solids
• Chapter 43 Nuclear Physics
• Chapter 44 Energy from the Nucleus
• Chapter 45 Quarks, Leptons, and the Big Bang