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lect136_1_w09revised

lect136_1_w09revised - Monday January 5 Lecture 1 Review...

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Monday January 5 Lecture 1 : Review : Vectors in R 2 and R 3 . (Refers to section 1.1 and 1.2 in your text) Objectives : 1. Define vectors in R n and in particular in R 2 and R 3 . 2. Define scalars, coordinates, components. 3. Represent vectors in R 2 and R 3 as directed line segments. 4. Add, subtract, scalar multiply, dot product R n . 5. Add, subtract, scalar multiply vectors when represented by directed line segments. 6. Recognize equivalent vectors, collinear, parallel vectors. 7. Define a linear combination of vectors, unit vector, midpoint of two points or vectors in R 2 and R 3 . 8. Give the standard basis of vectors for R 3 . 1.1 Introduction The elements, x = ( x 1 , x 2 ), of the Cartesian plane R 2 are familiar and usually referred to as ordered pairs , or coordinates of the plane. In this course we will begin referring to these as members of a larger family of mathematical objects called vectors . Observe how we represent vectors by a letter in boldface , while scalars are simply expressed in italics . 1.1.1 Definition Let R n = {( x 1 , x 2 , ..., x n ) : x 1 , x 2 , ..., x n R }. We say that x is an n-tuple of real numbers and we call x a vector . The x i 's are called the entries , coordinates or components of the vector. To distinguish vectors from scalars, we represent vectors in boldface. If we write x is a vector in R 5 , we mean x = ( x 1 , x 2 , x 3 , x 4 , x 5 ) where x 1 , x 2 , x 3 , x 4 and x 5 are real numbers. It is also common practice, whenever it useful to do so, to refer to a vector in R 2 and R 3 as a point . Hence to say, "let ( 1, 2, 0) be a point in R 3 " means the same thing as saying "let ( 1, 2, 0) be a vector in R 3 ". Occasionally we may refer to R 2 as the xy -plane and denote an arbitrary vector in R 2 by ( x , y ). Whenever we find it useful we might refer to R 3 as 3-space or the xyz -space and denote an arbitrary vector in R 3 by ( x , y , z ) instead of ( x 1 , x 2 , x 3 ).
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1.1.2 Remark We usually write vectors in the form x = ( x 1 , x 2 , ..., x n ) as above. But we could also write x as a column, x 1 x = x 2 : x 5 or as a diagonal if we wanted to. The important thing is that the order of the components is respected. If the vector is written horizontally we often specify this as say it is a " row vector ". Operation What we start with : What we get : Addition 2 vectors x and y in R n : x = ( x 1 , x 2 , ..., x n ) and y = ( y 1 , y 2 , ..., y n ). 1 vector x + y in R n : x + y = ( x 1 + y 1 , x 2 + y 2 , ..., x n + y n ) Scalar multiplication 1 vector x in R n : x = ( x 1 , x 2 , ..., x n ) and one number c in R (called a scalar ) 1 vector c x in R n : c x = ( cx 1 , cx 2 , ..., cx n ). Dot product 2 vectors x and y in R n : x = ( x 1 , x 2 , ..., x n ) and y = ( y 1 , y 2 , ..., y n ). 1 number < x , y > in R : < x , y > = x 1 y 1 + x 2 y 2 + ,...,+ x n y n . Other notation for dot-product: x y = < x , y > If it is written vertically, we will specify that it is a " column vector "
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1.2 Operations with vectors in R n .
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