Transcript (i) v - mlgibbons

A vector v is determined by two points in the plane: an
initial point P (also called the “tail” or basepoint) and a
terminal point Q (also called the “head”). We write
v  PQ
The length or magnitude of v, denoted v is the distance
Pythagorean THM or Distance Formula
from P to Q.
v  34
Q
34
P
5
3
v  OR is called the position
vector of R   3,5  .
The tail of a position vector is always the origin.
Vector terminology:
Two vectors v and w of nonzero length are called parallel if
the lines through v and w are parallel. Parallel vectors
point either in the same or in opposite directions.
A vector v is said to undergo a translation when it is
moved parallel to itself without changing its length or
direction. The resulting vector w is called a translate of v.
Translates have the same length and direction but
m2
different basepoints.
m1  m2  m3
m1
In many situations, it is convenient to treat
vectors with the same length and direction
as equivalent, even if they have different
basepoints. With this in mind, we say that
v and w are equivalent if w is a translate of v.
m3
P
Q
Every vector can be translated so that its tail is at the origin.
Therefore,
Every vector v is equivalent to a unique vector v0 based
at the origin.
Components of a vector (needed to work algebraically):
v  PQ, where P = (a1, b1) and Q = (a2, b2), are the quantities
a  a2  a1 (x-component), b  b2  b1 (y -component).
The pair of components is denoted a, b .
The length of a vector in terms of its
components (by the distance formula) is
v  PQ  a 2  b 2
The zero vector (whose head and
tail coincide) is the vector 0  0, 0 .
The vectors v and v0 have components a, b .
The components a, b determine the length and direction of
v, but not its basepoint. Therefore, two vectors have the
same components iff they are equivalent. Nevertheless, the
standard practice is to describe a vector by its components,
and thus we write
v  a, b
Although this notation is ambiguous (because it does not
specify the basepoint), it rarely causes confusion in practice.
To further avoid confusion, the following convention will be in
force for the remainder of the text:
All vectors are based at the origin, unless otherwise stated.
Determine whether v1  PQ
1 1 & v 2  P2Q2 are equivalent,
where
P1   3,7  , Q1   6,5 and P2   1, 4  , Q2   2,1
What is the magnitude of v1?
v1 = 6  3,5  7 = 3, 2 , v 2  2  1,1  4  3, 3
v1 and v2 are not equivalent
v1 = 3   2   13
2
2
We now define two basic vector operations:
Vector Addition and Scalar Multiplication.
The vector sum v + w is defined when v and w have
the same basepoint:
Without components, we use diagrams to describe these operations.
Translate w to the equivalent vector w’ whose tail
coincides with the head of v.
The sum v + w is the vector pointing from the tail of v
to the head of w’ .
If we had components, we use them to add,
subtract,...
We now define two basic vector operations:
Vector Addition and Scalar Multiplication.
Alternatively, we can use the Parallelogram Law:
v + w is the vector pointing from the basepoint to the
opposite vertex of the parallelogram formed by v and
w.
To add several vectors, translate so that they lie head
The vectors are adjacent sides of a parallelogram.
to tail.
To add several vectors, translate so that they lie
head to tail.
y
v '4
v '3
v
v '2
x
v '1
The sum v = v1 + v2 + v3 + v4.
Vector subtraction v − w is carried out by adding −w to v.
Or, more simply, draw the vector pointing from w to v, and
translate it back to the basepoint to obtain v − w.
v
vw
vw
w
w
v
w
Vector addition and scalar multiplication operations are easily
performed using components.
To add or subtract two vectors v and w, we add or subtract
their components.
 a  c, b  d
Similarly, to multiply v by a scalar λ, we multiply the
components of v by λ.
If λ > 0, λv points in the same direction as
a, b and in the opposite direction if λ < 0.
Vectors v and 2v are
based at P but 2v is
twice as long. Vectors v
and −v have the same
length but opposite
directions.
v  a, b is nonzero   a, b has length  v
Vector Operations Using Components
v  a, b and w  c, d 
(i) v + w = a  c, b  d
(ii) v − w = a  c, b  d
(iii) λv =  a,  b
P  1, 4 
Q   3, 2 
(iv) v + 0 = 0 + v = v
We also note that if P = (a1, b1) and Q = (a2, b2), then
components of the vector v  PQ are conveniently
computed as the difference
PQ  2, 2
PQ  a2  a1 , b2  b1
v  1, 4 and w  3, 2 , calculate v  w and 5v.
4, 6 and 5, 20
THEOREM 1 Basic Properties of Vector Algebra
For all vectors u, v, w and for all scalars λ,
Express u  4, 4 as a linear combination of
v  6, 2 and w  2, 4 .
y
Algebra I
Subtract & Substitute
u  4, 4 
A linear combination of vectors v and w is a
w  2, 4
vector rv + sw where r and s are scalars. If v and
w are not parallel, then every vector u in the
4
w
5
plane can be expressed as a linear combination
u = rv + sw.
v
w
2
4
6, 2  2, 4
5
5
v  6, 2
2
v
5
x
u
r 6, 2  s 2, 4  6r  2 s, 2r  4 s  4, 4
 6r  2 s  4
Parallelogram Law
2r  4 s  4
 4r  2 s  0  s  2r  10r  4  r  2 / 5
 s  2r  4 / 5
2
4
u  6, 2  2, 4
5
5
A vector of length 1 is called a unit vector. Unit vectors are
often used to indicate direction, when it is not necessary to
specify length. The head of a unit vector e (based at the
origin) lies on the unit circle and has components
e  cos  ,sin 
where θ is the angle between e and the positive x-axis.
CV
The head of a unit vector lies on the unit circle.
We can always scale a
nonzero vector v = a, b
to obtain a unit vector
pointing in the same
direction:
1
ev 
v
y
v  a, b
b
`
ev
v
e v  cos  ,sin 

1
a
x
v  a, b makes an angle 
with the positive x-axis 
ev  Unit vector in the direction of v.
v  a, b  v e v  v cos  ,sin 
To define ev , let's get back to v.
Find the unit vector in the direction of v = 3,5 .
1
ev 
v
34
3
5
,
34 34
34
5
3
It is customary to introduce a special notation for the unit
vectors in the direction of the positive x- and y-axes:
i  1, 0 ,
j  0,1
The vectors i and j are called the standard basis vectors.
Every vector in the plane is a linear combination of i and j:
v  a , b  ai  b j
1
ev 
v
v
*** ai  a
For example, 4, 2  4i  2 j. Vector addition is performed
by adding the i and j coefficients. For example,
j  0,1
i  1, 0
4i
x
v
2 j
y
v  4i  2 j
 4, 2 
(4i − 2j) + (5i + 7j) = (4 + 5)i + (−2 + 7)j = 9i + 5j
7j
 5, 7 
w  5i  7 j
w
j  0,1
i  1, 0
 9,5
v  w  9i  5j
vw
5i
x
v
y
v  4i  2 j
 4, 2 
CONCEPTUAL INSIGHT It is often said that quantities such as
force and velocity are vectors because they have both
magnitude and direction, but there is more to this statement
than meets the eye. A vector quantity must obey the law of
vector addition (Figure 18), so if we say that force is a vector,
we are really claiming that forces add according to the
Parallelogram Law. In other words, if forces F1 and F2 act on
an object, then the resultant force is the vector sum F1 + F2.
This is a physical fact that must be verified experimentally. It
was well known to scientists and engineers long before the
vector concept was introduced formally in the 1800s.
When an airplane traveling with velocity v1 encounters a wind of
velocity v2, its resultant velocity is the vector sum v1 + v2.
THEOREM 2 Triangle Inequality For any two vectors v and w,
vw  v + w
Equality holds only if v = 0 or w = 0, or if w = λv, where λ ≥ 0.