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Lecture

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University of Waterloo

Mathematics

MATH 136

Robert Andre

Spring

Description

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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
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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.
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- 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.
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- 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
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thing as saying "let (−1, 2, 0) be a vector in ℝ ".
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- 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
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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
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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".
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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 .
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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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