Lecture 3: Vectors - Dot Products - Cross Products - 3D Kinematics
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1. Vectors - Direction Distinguishes Vectors from Scalars
2. Decomposition of a Vector:
A vector can be projected onto three coordinate axes x,y,z, along which lie unit vectors (denoted with roofs). Professor Lewin works an example.
3. Scalar Product:
The "dot" product of two vectors is a scalar. A scalar can be positive, negative or zero and we'll use it later in the course to calculate work and energy. Professor Lewin calculates "A dot B" in a couple of examples.
4. Vector Product:
The cross product (also called vector product) of two vectors results in a vector. Professor Lewin presents two methods for calculating it. A cross product of the vectors A and B is always perpendicular to both A and B. The direction is easily found using the right-hand corkscrew rule. We'll use cross products to calculate torques and angular momentum later in the course. Always use Right Handed coordinate systems, x-hat cross y-hat gives z-hat. If you don't, you'll get into trouble for which you will have to pay dearly.
5. Decomposition of 3D Vectors r, v and a:
Professor Lewin writes the equations for position (r), velocity (v) and acceleration (a) showing their projection onto the x,y,z axes, and he introduces a shorthand notation for time derivatives. 3D motion can be reduced to three 1D motions which can greatly simplify matters.
6. Projectile Motion in the Vertical Plane:
Professor Lewin throws an object up, and decomposes its initial velocity into a horizontal and a vertical direction. If air drag can be ignored, the horizontal velocity remains constant. Gravitational acceleration is only in the vertical direction and is not affected by the horizontal motion. This acceleration is constant in the lecture hall if air drag can be ignored (see Lecture 12).
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