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Lesson 06: Vector Product of Vectors

Lesson 17/177 | Study Time: 30 Min

Course: Physics XI

Lesson 06: Vector Product of Vectors

Learning Outcomes

i. Students will be able to define the concept of the vector product of vectors.

ii. Students will be able to explain the significance of the vector product as a cross product.

iii. Students will be able to derive the formula for the vector product of two vectors.

iv. Students will be able to apply the vector product concept to determine the direction and magnitude of the resultant vector.

Introduction

In the realm of vectors, the vector product, also known as the cross product, stands as another fundamental concept alongside the scalar product. While the scalar product results in a scalar quantity, the vector product yields a vector quantity, meaning a quantity with both magnitude and direction. This vector product provides valuable insights into the relationship between two vectors, particularly their relative orientation in three-dimensional space.

i. Understanding the Vector Product as a Cross Product

The vector product can be visualized as the perpendicular vector arising from the interaction of two vectors. Imagine two vectors, A and B, positioned in three-dimensional space. The vector product of A and B, denoted by A × B, is a vector perpendicular to both A and B, lying in a plane defined by the two original vectors.

ii. Deriving the Formula for the Vector Product

The formula for the vector product of two vectors, A and B, is given by:

A × B = |A| |B| sin(θ) n

where:

|A| and |B| represent the magnitudes of vectors A and B, respectively

θ is the angle between vectors A and B (less than 180 degrees)

n is a unit vector perpendicular to both A and B, pointing in the direction determined by the right-hand rule

This formula highlights the relationship between the vector product, the magnitudes of the vectors, the angle between them, and the direction of the resultant vector.

Applying the Vector Product to Determine Direction and Magnitude

The vector product can be used to determine the direction and magnitude of the resultant vector. The direction of the resultant vector is perpendicular to both the original vectors and follows the right-hand rule. The magnitude of the resultant vector is given by:

Magnitude of resultant vector = |A| |B| sin(θ)

This allows us to quantify the relative orientation and strength of the interaction between the two vectors.

The vector product of vectors plays a crucial role in various fields of physics and engineering. Its ability to represent the perpendicular vector arising from two vectors and determine its direction and magnitude has wide-ranging applications. By understanding the vector product and its formula, students gain a deeper understanding of vector interactions and their significance in various physical scenarios.

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Class Sessions

1- Lesson 01: The Role of Physics in Science, Technology, and Society 2- Lesson 02: SI Units: Base, Derived, and Supplementary 3- Lesson 03: Expressing Derived Units from Base Units 4- Lesson 04: SI Unit Conventions 5- Lesson 05: Uncertainty in Measurements 6- Lesson 06: Least Count and Resolution of Measuring Instruments 7- Lesson 07: Precision versus Accuracy 8- Lesson 08: Uncertainty in Derived Quantities 9- Lesson 09: Scientific Notation, Significant Figures, and Units in Numerical and Practical Work 10- Lesson 10: Dimensionality and Homogeneity of Physical Equations 11- Lesson 11: Deriving Formulas Using Dimensions 12- Lesson 01: The Cartesian Coordinate System 13- Lesson 02: Vector Addition Using Head-to-Tail Rule 14- Lesson 03: Resolving Vectors into Perpendicular Components 15- Lesson 04: Vector Addition Using Perpendicular Components 16- Lesson 05: Scalar Product of Vectors 17- Lesson 06: Vector Product of Vectors 18- Lesson 07: Direction of Vector Product 19- Lesson 08: Torque as Vector Product 20- Lesson 09: Applications of Torque 21- Lesson 10: First Condition of Equilibrium 22- Lesson 11: Second Condition of Equilibrium 23- Lesson 12: Solving Two-Dimensional Equilibrium Problems 24- Lesson 01: Vector Nature of Displacement 25- Lesson 02: Average and Instantaneous Velocities 26- Lesson 03: Comparison of Speeds and Velocities 27- Lesson 04: Interpreting Displacement-Time and Velocity-Time Graphs 28- Lesson 05: Determining Instantaneous Velocity from Displacement-Time Graph 29- Lesson 06: Average and Instantaneous Acceleration 30- Lesson 07: Positive, Negative, Uniform, and Variable Acceleration 31- Lesson 08: Determining Instantaneous Acceleration from Velocity-Time Graph 32- Lesson 09: Manipulating Equations of Uniformly Accelerated Motion 33- Lesson 10: Projectile Motion as Two-Dimensional Motion 34- Lesson 11: Projectile Motion in Absence of Air Resistance 35- Lesson 12: Horizontal Component of Velocity in Projectile Motion 36- Lesson 13: Acceleration in Projectile Motion 37- Lesson 14: Independence of Horizontal and Vertical Motions 38- Lesson 15: Analyzing Projectile Motion Using Equations 39- Lesson 16: Projectile Launched from Ground Height 40- Lesson 17: Angle for Maximum Range Height 41- Lesson 18: Launch Angles for Same Range 42- Lesson 19: Effect of Air Resistance on Projectile Motion 43- Lesson 20: Applying Newton's Laws to Object Motion 44- Lesson 21: Mass as a Property of Matter 45- Lesson 22: Weight as an Effect of Gravity 46- Lesson 23: Newton's Second Law of Motion as Rate of Change of Momentum 47- Lesson 24: Newton's Third Law of Motion and Conservation of Momentum 48- Lesson 25: Limitations of Newton's Laws of Motion 49- Lesson 26: Impulse as a Product of Impulsive Force and Time 50- Lesson 27: Effect of Impulsive Force on Momentum 51- Lesson 28: Conservation of Momentum in Collisions 52- Lesson 29: Solving Problems of Elastic and Inelastic Collisions 53- Lesson 30: Momentum Conservation in All Situations 54- Lesson 31: Relative Speed of Approach and Separation in Elastic Collisions 55- Lesson 32: Distinction Between Explosion and Collision 56- Lesson 01: Concept of Work 57- Lesson 02: Positive, Negative, and Zero Work 58- Lesson 03: Calculating Work from Force-Displacement Graph 59- Lesson 04: Gravitational Field as an Example of Field of Force 60- Lesson 05: Proving Gravity as a Conservative Field 61- Lesson 06: Work Done by Gravity 62- Lesson 07: Gravitational PE and Reference Level 63- Lesson 08: Potential at a Point 64- Lesson 09: Escape Velocity 65- Lesson 10: Conservative vs. Non-Conservative Forces 66- Lesson 11: Power as Scalar Product of Force and Velocity 67- Lesson 12: Work Done Against Friction and Dissipation of Heat 68- Lesson 13: Energy Losses, Efficiency, and Practical Devices 69- Lesson 14: Work-Energy Theorem in Resistive Medium 70- Lesson 15: Limitations of Conventional Sources of Energy 71- Lesson 16: Potentials of Non-Conventional Sources of Energy 72- Lesson 01: Angular Displacement, Angular Velocity, and Angular Acceleration 73- Lesson 02: Solving Problems Using S = rθ and v = rω 74- Lesson 03: Using Equations of Angular Motion to Solve Problems 75- Lesson 04: Qualitative Description of Motion in a Curved Path 76- Lesson 05: Deriving and Using Centripetal Acceleration 77- Lesson 06: Solving Problems Using Centripetal Force 78- Lesson 07: Situations Involving Centripetal Acceleration 79- Lesson 08: Banking Angle and Vehicle Speed 80- Lesson 09: Relating Banking Angle to Vehicle Speed and Radius of Curvature 81- Lesson 10: Satellite Orbits and Gravitational Force 82- Lesson 11: Weightlessness in Orbiting Satellites 83- Lesson 12: Creating Artificial Gravity 84- Lesson 13: Orbital Velocity and Its Relationship to Gravitational Force 85- Lesson 14: Applications of Satellites 86- Lesson 15: Communication Satellites and Their Orbits 87- Lesson 16: Moment of Inertia and Angular Momentum 88- Lesson 17: Torque, Moment of Inertia, and Angular Acceleration 89- Lesson 18: Conservation of Angular Momentum 90- Lesson 19: Solving Problems Using Formulas for Moment of Inertia 91- Lesson 01: Characteristics of Ideal Fluid Flow 92- Lesson 02: Transition from Laminar to Turbulent Flow 93- Lesson 03: Prevalence of Turbulent Flow 94- Lesson 04: Equation of Continuity for Ideal and Incompressible Fluids 95- Lesson 05: Connection of Equation of Continuity to Conservation of Mass 96- Lesson 06: Pressure Difference and Bernoulli Effect 97- Lesson 08: Applications of Bernoulli's Effect 98- Lesson 07: Deriving Bernoulli's Equation for Horizontal Tube Flow 99- Lesson 09: Real Fluids and Viscosity 100- Lesson 10: Viscous Forces and Retarding Force 101- Lesson 11: Dependence of Viscous Force on Shape and Velocity 102- Lesson 12: Dimensional Analysis and Stokes' Law 103- Lesson 13: Terminal Velocity of a Spherical Body 104- Lesson 01: Simple Examples of Free Oscillations 105- Lesson 02: Conditions for Simple Harmonic Motion (SHM) 106- Lesson 03: SHM from Circular Motion 107- Lesson 04: Defining SHM Parameters 108- Lesson 05: Defining Equation of SHM 109- Lesson 06: Proving SHM of Mass-Spring System 110- Lesson 07: Energy Exchange in SHM 111- Lesson 08: SHM of a Simple Pendulum 112- Lesson 09: Practical Examples of Oscillations 113- Lesson 10: Graphical Representation of Forced Oscillation 114- Lesson 11: Damped Oscillations and Critical Damping 115- Lesson 01: Introduction to Wave Motion 116- Lesson 02: Mechanical Waves vs. Electromagnetic Waves 117- Lesson 03: Key Terms in Wave Model 118- Lesson 04: Solving Problems Using v = fλ 119- Lesson 05: Energy Transfer in Progressive Waves 120- Lesson 06: Characteristics of Sound Waves 121- Lesson 07: Transverse vs. Longitudinal Waves 122- Lesson 08: Speed of Sound and Newton's Formula 123- Lesson 09: Laplace Correction in Newton's Formula 124- Lesson 10: Factors Affecting Speed of Sound in Air 125- Lesson 11: Superposition of Waves 126- Lesson 12: Interference of Sound Waves 127- Lesson 13: Beats Formation Due to Interference 128- Lesson 14: Formation of Stationary Waves 129- Lesson 15: Nodes and Antinodes 130- Lesson 16: Modes of Vibration of Strings 131- Lesson 17: Stationary Waves in Vibrating Air Columns 132- Lesson 18: Doppler Effect in Mechanical Waves 133- Lesson 19: Doppler Effect in Electromagnetic Waves 134- Lesson 20: Generation and Detection of Ultrasonic Waves 135- Lesson 21: Ultrasound for Diagnostic Information 136- Lesson 01: Light Waves in Electromagnetic Spectrum 137- Lesson 02: Wave Fronts 138- Lesson 03: Huygen's Principle and Wave Front Construction 139- Lesson 04: Conditions for Interference 140- Lesson 05: Young's Double Slit Experiment 141- Lesson 06: Color Patterns in Thin Films 142- Lesson 07: Michelson Interferometer and Its Applications 143- Lesson 08: Diffraction and Interference 144- Lesson 09: Diffraction as Evidence of Light's Wave Nature 145- Lesson 10: Diffraction at a Narrow Slit 146- Lesson 11: Diffraction Grating for Wavelength Determination 147- Lesson 12: Diffraction of X-rays through Crystals 148- Lesson 13: Polarization as a Transverse Wave Phenomenon 149- Lesson 14: Polarization by Polaroid 150- Lesson 15: Effect of Polaroid Rotation on Polarization 151- Lesson 16: Production and Detection of Plane Polarized Light 152- Lesson 01: Thermal Energy Transfer 153- Lesson 02: Thermal Equilibrium 154- Lesson 03: Heat Flow and Work as Energy Transfer 155- Lesson 04: Thermodynamics and Related Terms 156- Lesson 05: Rise in Temperature and Internal Energy 157- Lesson 06: Mechanical Equivalent of Heat 158- Lesson 07: Internal Energy and Its Determination 159- Lesson 08: Work Done by a Thermodynamic System 160- Lesson 09: First Law of Thermodynamics 161- Lesson 10: Conservation of Energy 162- Lesson 11: Defining Specific Heat and Molar Specific Heat 163- Lesson 12: Deriving Cp - Cv = R from the First Law of Thermodynamics 164- Lesson 13: Understanding the Working Principle of Heat Engines 165- Lesson 14: Differentiating Reversible and Irreversible Processes 166- Lesson 15: Comprehending the Second Law of Thermodynamics 167- Lesson 16: Exploring the Working Principle of Carnot's Engine 168- Lesson 17: Comparing Refrigerators to Heat Engines 169- Lesson 18: Applying Specific Heat and Molar Specific Heat Concepts 170- Lesson 19: Deriving the Coefficient of Performance for Refrigerators 171- Lesson 20: Understanding the Relationship between Entropy Change and Heat Transfer 172- Lesson 21: Deriving the Coefficient of Performance (COP) of Refrigerators 173- Lesson 22: Understanding Entropy Change and Heat Transfer 174- Lesson 23: Connecting Temperature Increase to Disorder 175- Lesson 24: Recognizing Energy Degradation as Entropy Increase 176- Lesson 25: Understanding Energy Degradation in Natural Processes 177- Lesson 26: Identifying the Tendency Towards Disorder