2
Geometry of R and R
3
Vectors in R2 and R3
NOTATION

R The set of real numbers

R2 The set of ordered pairs of real numbers

R3 The set of ordered triples of real numbers
Vector



A vector in R2 (or R3) is a directed line
segment from the origin to any point in R2 (or
R3)
Vectors in R2 are represented using ordered
pairs
Vectors in R3 are represented using ordered
triples
Notation for Vectors



Vectors in R2 (or R3) are denoted using bold
faced, lower case, English letters
Vectors in R2 (or R3) are written with an arrow
above lower case, English letters
Points in R2 (or R3) are denoted using upper
case English letters
Example 1

u = (u1, u2, u3) represent a vector in R3 from the
origin to the point P (u1, u2, u3)

u1, u2, and u3 are the components of the u
Equality of Two Vectors



Two vectors are equal if their corresponding
components are equal.
That is, u = (u1, u2, u3) and v = (v1, v2, v3) are
equal if and only if u1 = v1, u2 = v2, and u3 = v3
Hence, if u = 0, the zero vector, then u1 = u2 =
u3 = 0.
Collinear Vectors


Two vectors are collinear if thy both lie on the
same line.
That is, u = (u1, u2, u3) and v = (v1, v2, v3) are
collinear if the points U, V, and the Origin are
collinear points.
2
Length of a Vector in R
The length (norm, magnitude) of v = (v1, v2),
denoted by ||v||, is the distance of the
point V (v1, v2) from the origin.
v  v1  v2
2
2
3
Length of a Vector in R
The length (norm, magnitude) of v = (v1, v2,
v3) is the distance of the point V (v1, v2, v3)
from the origin.
v  v1  v2  v3
2
2
2
Example
Find the length of u = (-4, 3, -7)
u  (4)  (3)  (7)
2
 74
2
2
Zero Vector and Unit Vector


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The magnitude of 0 is zero.
If a vector has length zero, then it is 0
If a vector has magnitude 1, it is called a
unit vector.
Scalar Multiplication
Let c be a scalar and u a vector in R2 (or R3).
Then the scalar multiple of u by c is the
vector the vector obtained by multiplying
each component of u by c.
That is, cu = (cu1, cu2) in R2, and
cu = (cu1, cu2, cu3) in R3
Example
Find cu for u = (-4, 0, 5) and c = 2.
If v = (-1, 1), sketch v, 2v and -2v.
Theorem 1.1.1
Let u be a nonzero vector in R2 or R3, and c be
any scalar. Then u and cu are collinear, and
a) if c > 0, then u and cu have the same
direction
b) if c < 0, then u and cu have opposite
directions
c) ||cu|| = |c| ||u||
Example
Let u = (-4, 8, -6)
a) Find the midpoint of the vector u.
b) Find a the unit vector in the direction of u.
c) Find a vector in the direction opposite to u
that is 1.5 times the length of u.
Vector Addition
Let u and v be nonzero vectors in R2 or R3.
Then the sum u + v is obtained by adding
the corresponding components.
That is,
2
 u + v = (u1 + v1, u2 + v2), in R
3
 u + v = (u1 + v1, u2 + v2, u3 + v3), in R
Example
Find the sum of each pair of vectors
1.
2.
u = (2, 1, 0) and v = (-1, 3, 4)
u = (1, -2) and v = (-2, 3)
Sketch each vector in part (2) and their sum.
Theorem 1.1.2
For nonzero vectors u and v the directed line
segment from the end point of u to the
endpoint of u + v is parallel and equal in
length of v.
Proof of Theorem 2: Outline
1)
2)
3)
4)
5)
Show that d(u, u+v) = d(0, v).
Show that d(v, u+v) = d(0, u).
The above two parts proves that the four line
segments form a parallelogram.
The opposite sides of a parallelogram are parallel and
of the same length. (A result from Geometry.)
We must also prove that the four vectors u, v, u + v,
and 0 are coplanar, which will be done in section 1.2.
Opposite and Vector Subtraction
Let u be vector in R2 or R3. Then
1. Opposite or Negative of u, denoted by –u,
is (-1)(u).
2. The difference u – v is defined as u +(–v).
Theorem 1.1.3
Let u, v and w be vectors in R2 or R3, and c
and d scalars. Then
1. u + v = v + u
2. (u + v) + w = v + (u + w)
3. u + 0 = u
4. u + (-u) = 0
5. (cd)u = c(du)
Theorem 3 Cont’d.
Let u, v and w be vectors in R2 or R3, and c
and d scalars. Then
6. (c + d)u = cu + du
7. c(u + v) = cu + cv
8. 1u = u
9. (-1)u = -u
10. 0u = 0
Equivalent Directed Line Segments
Two directed line segments are said to be
equivalent if they have the same direction
and length.
Theorem 1.1.4
Let U and V be distinct points in R2 or R3. Then the
vector v – u is equivalent to the directed line
segment from U to V. That is,
1. The line UV is parallel to the vector v – u, and
2. d(u, v) = ||v – u||
Proof of Theorem 4: Outline
1.
2.
Show that the sum of u and v – u is v.
This proves that the two vectors v – u is
parallel and equal in length to the directed line
segment from U to V.
Example
Is the line determined by (3,1,2) & (4,3,1), parallel
to the line determined by (1,3,-3) & (-1,-1,-1)?
Outline for the solution: Find unit vectors in the
direction of the lines. If they are same or
opposite, then the two vectors are parallel.
Standard Basis Vectors in R
i = (1, 0)
j = (0, 1)
If (a, b) is a vector in R2, then
(a, b) = a(1, 0) + b(0, 1) = ai + bj.
2
Standard Basis Vectors in R
i = (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)
If (a, b, c) is a vector in R3, then
(a, b, c) = ai + bj + ck
3
Example
Express (2, 0, -3) in i, j, k form.
Homework 1.1