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MATH 136 (145)
Lecture

lect136_1_w14.pdf

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School
University of Waterloo
Department
Mathematics
Course
MATH 136
Professor
Robert Andre
Semester
Spring

Description
2 3 Monday, January 6 − Lecture 1 : Vectors in ℝ and ℝ . Concepts: 1. Vectors in ℝ n, in ℝ and ℝ . 3 2. Scalars, coordinates, components. 3. Addition and scalar multiplication of vectors. 4. Linear combinations of vectors 5. Collinear vectors 6. Line, its vector equation 7. Direction vector of a line 1.1 Vectors − The elements, x = (x , x ), of the Cartesian plane ℝ are familiar and 1 2 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. n 1.1.1 Definition − Let ℝ = {(x , x , 1..,2x ) : xn, x , 1..,2x ∈ ℝ}n We say that x is an n-tuple of real numbers and we call x avector. The x's are callid the entries, coordinates or components of the vector. To distinguish vectors from scalars, we represent vectors in boldface. 5 - If we write x is a vector in ℝ , we mean x = (x , 1 , 2 , 3 , 4 )5where x , x1, x2, x3 4 and x 5re real numbers. 2 - It is also common practice, whenever it useful to do so, to refer to a vector in ℝ and ℝ as a point. Hence to say, "let (−1, 2, 0) be a point in ℝ " means the same 3 thing as saying "let (−1, 2, 0) be a vector in ℝ ". 2 - Occasionally we may refer to ℝ as the xy-plane and denote an arbitrary vector in ℝ by (x, y). Whenever we find it useful we might refer to ℝ as “3-space” or the 3 xyz-space and denote an arbitrary vector in ℝ by (x, y, z) instead of (x , x , x 1. 2 3 1.1.2 Remark − We can write vectors in the form x = (x , x , ..., x ) as above. But we 1 2 n could also write x as a column, or as a diagonal if we wanted to. - The important thing is that the order of the components is respected. - If a vector is written vertically, some refer to it as "column vector" - If the vector is written horizontally we often specify this by saying it is a "row vector". n 1.2 Operations with vectors in ℝ . Operation What we start with : What we get : 2 vectors x and y in ℝ : n n 1 vector x + y in ℝ : Addition x = (x 1 x2, ..., n ) x + y and y = ( y 1 y 2 ..., n ). = (x + y , x + y , ..., x + y ) 1 1 2 2 n n n n 1 vector x in ℝ : 1 vector cx in ℝ : Scalar x = (x1, x2, ...,nx ) and cx = (cx , cx , ..., cx ). multiplication one number c in ℝ 1 2 n (cx is called a scalar multiple of x) (called a scalar) 1.2.1 Examples − If x = (2, 1, 7) and y = (3, − 9, 0) 1. verify that 3x + 2y = (12, −15, 21). 2. verify that 3x − y = (3, 12, 21). 1.2.2 Equality of vectors: Two vectors a = (a , a , a1) a2d b3= (b , b , b 1 ar2 sa3d to be equal vectors if and only if their corresponding entries are equal, that is, a 1 = b 1 a 2 b and a = b . 2 3 3 1.2.3 Definition − Two vectors a and b are said to be collinear if one vector is a scalar multiple of the other. Essentially this means that these two points lie on a single line going through the origin. 1.2.4 Definition − A linear combinations of vectors is a sum of scalar multiples of vectors. Example − Witness (2, 3) = 2(1, 0) + 3(0, 1). So is a linear combination of the two vectors (1, 0) and (0, 1). More generally for any vector (a, b), (a, b) = a(1, 0) + b(0, 1) So every vector (a, b) in ℝ is a linear combination of the two vectors e1 = (1, 0) e2= (0, 1) 1.2.5 Properties of + an
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