## Lecture 3: Vectors - Dot Products - Cross Products - 3D Kinematics

recorded by: Massachusetts Institute of Technology, MIT

published: Oct. 10, 2008, recorded: September 1999, views: 95763

released under terms of: Creative Commons Attribution Non-Commercial Share Alike (CC-BY-NC-SA)

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# Description

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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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## Reviews and comments:

akpedye michael ufuoma, March 24, 2009 at 9:58 p.m.:very nice lecture...i have really understood some basic facts

Justin Van Horne, September 20, 2009 at 7:54 p.m.:That was a great lecture -- Awesome review material.

pre-medical student, September 5, 2010 at 12:07 p.m.:this lecture is so good

zeeshan, April 5, 2011 at 6:17 a.m.:sir please tell me why we take dot product as cos and cross product as sin

Godfred, October 7, 2011 at 4:04 p.m.:How to get these lectures on DVD pack?

Jm, December 2, 2011 at 3:19 a.m.:Clear course. Thanks for sharing!

Two little corrections maybe about cross product. It always gives birth to a vector, never a scalar!

1/ So when it's null (because the 2 vectors are orthogonal) we got the null vector 0-> (sorry for the bad notation, there's no latex here...) not 0 (scalar)!

2/ Same thing for method II: A-> x B-> = |A|.|B|.sin(theta).|A-> x B->|/(|A|.|B|)

Jm, December 2, 2011 at 3:24 a.m.:I wasn't right above... I'd have written:

2/ Same thing for method II: A-> x B-> = |A|.|B|.sin(theta).(A-> x B->)/|A-> x B->|

Well, what I simply wanted to say was that a vector was missing on the board... That's all!

Haider Ali, November 19, 2015 at 4:40 p.m.:Sir What Is 26.100 ????????????????????

Prem kumar, March 26, 2017 at 8:41 p.m.:Is it meaningful to add a component of a vector to the same vector?

Prem kumar, March 26, 2017 at 8:45 p.m.:Is component of a vector always scalar?

Davor form VideoLectures, December 18, 2017 at 10:34 a.m.:Hi all!

We have translated this entire course for you from English into 11 languages.

Check this video and give us some feedback in this short survey https://www.surveymonkey.co.uk/r/6DMBC3Q

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