18.02SC Notes: Vectors - MIT OpenCourseWare
VectorsOur very first topic is unusual in that we will start with a brief written presentation. Moretypically we will begin each topic with a videotaped lecture by Professor Auroux and followthat with a brief written presentation.As we pointed out in the introduction, vectors will be used throughout the course. Thebasic concepts are straightforward, but you will have to master some new terminology.Another important point we made earlier is that we can view vectors in two differentways: geometrically and algebraically. We will start with the geometric view and introduceterminology along the way.Geometric viewA vector is defined as having a magnitude and a direction. We represent it by an arrow inthe plane or in space. The length of the arrow is the vector’s magnitude and the directionof the arrow is the vector’s direction.In this way, two arrows with the same magnitude and direction represent the same vector.//////////(same vector)////////We will refer to the start of the arrow as the tail and the end as the tip or head. The vector between two points will be denoted PQ.// Q///// // PQ/// /P We call P the initial point and Q the terminal point of PQ.The magnitude of the vector A will be denoted A . Magnitude will also be called lengthor normScaling, adding and subtracting vectorsScaling a vector means changing its length by a scale factor. For example,//////////2A /////////////// A //A /////// 12 A //////////////////// ////(scaling a vector)Because we use numbers to scale a vector we will often refer to real numbers as scalarsYou add vectors by placing them head to tail. As the figure shows, this can be done in1
either order/////AA B///////BB/ //////A/////It is often useful to think of vectors as displacements. In this way, A B can be thoughtof as the displacement A followed by the displacement B.You subtract vectors either by placing the tail to tail or by adding A ( B). B////A////////A BBThought of as displacements A B is the displacement from the end of B to the end of A.Algebraic viewAs is conventional, we label the origin O. In the plane O (0, 0) and in space O (0, 0, 0).In the xy-plane if we place the tail of A at the origin, its head will be at the point withcoordinates, say, (a1 , a2 ). In this way, the coordinates of the head determine the vector A.When we draw A from the origin we will refer to it as an origin vector.Using the coordinates we writeA (a1 , a2 ).Addition, subtraction and scaling using coordinates is discussed below.Graphically:yy / y (a1 , a2 )////A///a2 j////////xOa1 iy The vectors i and j used in the figure above have coordinatesi (1, 0), j (0, 1). We use them so often that they gettheir own symbols.jiNotation and terminology1. (a1 , a2 ) indicates a point in the plane.2. (a1 , a2 ) a1 i a2 j. This is equal to the vector drawn from the origin to the point(a1 , a2 ).3. For A a1 i a2 j, a1 and a2 are called the i and j components of A. (Note that theyare scalars.) 5. P OP is the vector from the origin to P .2
6. On the blackboard vectors will usually have an arrow above the letter. In print we will often drop the arrow and just use the bold face to indicate a vector, i.e. P P .7. A real number is a scalar, you can use it to scale a vector.Vector algebra using coordinatesFor the vectors A a1 i a2 j and B b1 i b2 j we have the following algebraic rules. Thefigures below connect these rules to the geometric viewpoint.Magnitude: A a21 a22 (this is just the Pythagorean theorem)Addition: A B (a1 b1 )i (a2 b2 )j, that is, (a1 , a2 ) (b1 , b2 ) (a1 b1 , a2 b2 )Subtraction: A B (a1 b1 )i (a2 b2 )j, that is, (a1 , a2 ) (b1 , b2 ) (a1 b1 , a2 b2 )b1a1a1Bb2/////////A BA///a2//////////////a2 b2a2 b2Aa2A Ba1 b1Ba1 b1b2b1 For two points P and Q the vector PQ Q P i.e., PQ is the displacement from P to Q. Q (q1 , q2 ) PQy yQ P (p1 , p1 )PxOVectors in three dimensionsWe represent a three dimensional vector as an arrow in space. Using coordinates we needthree numbers to represent a vector.zy // (a1 , a2 , a3 )///A///////y A (a1 , a2 , a3 )xGeometrically nothing changes for vectors in three dimensions. They are scaled and addedexactly as above.3
Algebraically the origin vector A ha1 , a2 , a3 i starts at the origin and extends to the point(a1 , a2 , a3 ). We have the special vectors i h1, 0, 0i, j h0, 1, 0i, k h0, 0, 1i. Usingthemha1 , a2 , a3 i a1 i a2 j a3 k.Then, for A ha1 , a2 , a3 i and B hb1 , b2 , b3 i we haveha1 , a2 , a3 i hb1 , b2 , b3 i ha1 b1 , a2 b2 , a3 b3 i.exactly as in the two dimensional case.Magnitude in three dimensions also follows from the Pythagorean theorem.q a1 i a2 j a3 k ha1 , a2 , a3 i a21 a22 a23You can see this in the figure below, where r pa21 a22 and A ppr2 a23 a12 a22 a23 .zO? (a1 , a2 , a3 ) A a3 O OO/ytOOOttta1tr OOOttttta2xztUnit vectorsA unit vector is any vector with unit length. When we want to indicate that a vector is abunit vector we put a hat (circumflex) above it, e.g., u.The special vectors i, j and k are unit vectors.Since vectors can be scaled, any vector can be rescaled to be a unit vector.Example: Find a unit vector that is parallel to h3, 4i.13 4Answer: Since h3, 4i 5 the vector h3, 4i ,has unit length and is parallel to55 5h3, 4i.4
MIT OpenCourseWarehttp://ocw.mit.edu18.02SC Multivariable CalculusFall 2010For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Unit vectors A unit vector is any vector with unit length. When we want to indicate that a vector is a unit vector we put a hat (circum ex) above it, e.g., u. The special vectors i, j and k are unit vectors. Since vectors can be scaled, any vector can be rescaled b to be a unit vector. Example: Find a unit vector that is parallel to h3;4i. 1 3 4
I think of atomic vectors as “just the data” Atomic vectors are the building blocks for augmented vectors Augmented vectors Augmented vectors are atomic vectors with additional attributes attach
Two nonparallel vectors always define a plane, and the angle is the angle between the vectors measured in that plane. Note that if both a and b are unit vectors, then kakkbk 1, and ab cos . So, in general if you want to find the cosine of the angle between two vectors a and b, first compute the unit vectors aˆ and bˆ in the directions of a .
Draw vectors on your map from point to point along the trip through NYC in different colors. North Vectors-Red South Vectors-Blue East Vectors-Green West Vectors-Yellow Site Address Penn Station 33 rd St and 7th Ave Empire State Building 34th St and 5th Ave NY NY Library 41st and 5th Ave .
Chapter 6 139 Vectors and Scalars (ii) Vectors Addition is Associative: i.e. a b c a b c where . a , b . and . c . are any three vectors. (iii) O is the identity in vectors addition: Fig.9. For every vector . a O a Where . O. is the zero vector. Remarks: Non-parallel vectors are not added or subtracted by the .
6.1 An Introduction to Vectors, pp. 279-281 1. a.False. Two vectors with the same magnitude can have different directions, so they are not equal. b. True. Equal vectors have the same direction and the same magnitude. c. False. Equal or opposite vectors must be parallel and have the same magnitude. If two parallel vectors
Advanced Higher Notes (Unit 3) Vectors, Lines and Planes M Patel (April 2012) 1 St. Machar Academy Vectors, Lines and Planes Prerequisites: Adding, subtracting and scalar multiplying vectors; calculating angles between vectors. Maths Applications: Describing geometric transformations.
For two parallel vectors a b 0 4. The vector product of two vectors given in cartesian form We now consider how to find the vector product of two vectors when these vectors are given in cartesian form, for example as a 3i 2j 7k and b 5i 4j 3k where i, j and k are unit vectors in the directions of the x, y and z axes respectively.
Animal Food Fun & MORE. Instructions Equipment: Paper plate Thin card (not paper as it is too thin) Yellow and brown paint (or felt pen). Yellow bendy straws (you can colour paper ones) Sellotape Glue Elastic What to do: 1) Draw this shape on the back of your paper plate and cut it out carefully. (save this to make the ears). 2) Paint the front of both pieces of the .
