Download Vectors in R2 and R3: A Comprehensive Guide with Examples and Exercises and more Lecture notes Linear Algebra in PDF only on Docsity!
1. 1 Vectors in R
and R
Quote. “We must admit with humility that, while number is purely a product of our minds, space
has a reality outside our minds, so that we cannot completely prescribe its properties a priori” Carl
Friedrich Gauss ( 1777 - 1855 )
Vocabulary.
• R
2 : two-dimensional space.
• R
3 : three-dimensional space.
- n-tuple: an ordered list of n numbers enclosed in brackets.
- coordinates: the n-tuple that describes a position of a point.
- scalar: an object that can be described entirely by a number.
- vector: an object that can be described by a number value (called its magnitude) and a
direction. Can also be described by an n-tuple.
- components: each item of a vector.
- Graphical Representation of points and vectors in R
2 and R
3
- Notation for scalars, vectors, and points.
a, b, c, k, l, m, u, v, w for scalars
a, b, c, k, l, m, u, v, w or !a,
b, !c for vectors
- Addition and Scalar Multiplication of vectors
- Two vectors are equal if they have same magnitude and direction.
- If the length (also called the magnitude) of the vector is 0 then the direction does not
matter and such a vector is called the zero vector and denoted by 0.
vector addition
Geometric Algebraic
1
2
3
4
1 2 3 4
x
y
0
negative of a vector
Geometric Algebraic
1
2
3
4
1 2 3 4
x
y
0
scalar multiplication
Geometric Algebraic
1
2
3
4
1 2 3 4
x
y
0
vector subtraction
Geometric Algebraic
- Vector Equations of lines in R
2
Definition. A line through the origin in R
2 is a set of the form {t
d|t ∈ R} where
d is a
vector through the origin.
Example.
And then, how does the vector equation of the line through a point which is not the origin
look?
- Parametric Equations of lines in R
2
Definition. A through the origin in R
2 can be written parametrically by separating the
vector equation into two parametric equations. From the parametric equations, the parameter
can be elimination to give the familiar slope-intercept form of a line
Example.
- Vector and parametric equations of lines in R
3
We simply repeat what we have done in 2 dimensions with 2 component vectors and now
consider that we are in three dimensions and each vector has 3 components.
Example.
- Vector and parametric equations of planes in R
3
Definition. A plane through the origin in R
3 is a set of the form {t!u + s!v|s, t ∈ R} where
!u and !v are vectors through the origin.
Example.
CAUTION: if !v is a scalar multiple of !u then we have a line, not a plane!
