Transcript Algebra and Trigonometry

Vectors
Dr .Hayk Melikyan
Departmen of Mathematics and CS
[email protected]
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Directed Line Segments and Geometric Vectors
A line segment to which a direction has been assigned is called a directed line
segment. The figure below shows a directed line segment form P to Q. We call
P the initial point and Q the terminal point. We denote this directed line
segment by PQ.
Q
Initial point
Terminal point
P
The magnitude of the directed line segment PQ is its length. We
denote this by || PQ ||. Thus, || PQ || is the distance from point P to point Q.
Because distance is nonnegative, vectors do not have negative magnitudes.
Geometrically, a vector is a directed line segment. Vectors are often
denoted by a boldface letter, such as v. If a vector v has the same magnitude
and the same direction as the directed line segment PQ, we write
v = PQ.
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Vector Multiplication
If k is a real number and v a vector, the vector kv is
called a scalar multiple of the vector v. The
magnitude and direction of kv are given as follows:
The vector kv has a magnitude of |k| ||v||. We
describe this as the absolute value of k times the
magnitude of vector v.
The vector kv has a direction that is:
the same as the direction of v if k > 0, and
opposite the direction of v if k < 0
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The Geometric Method for Adding Two Vectors
A geometric method for adding two vectors is shown below. The sum of u + v
is called the resultant vector. Here is how we find this vector.
1. Position u and v so the terminal point of u extends from the initial
point of v.
2. The resultant vector, u + v, extends from the initial point of u to the
terminal point of v.
Resultant vector
u+v
v
Terminal point of v
u
Initial point of u
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The Geometric Method for the Difference of Two
Vectors
The difference of two vectors, v – u, is defined as v – u = v + (-u), where –u is
the scalar multiplication of u and –1: -1u. The difference v – u is shown below
geometrically.
-u
v–u
-u
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v
u
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The i and j Unit Vectors
Vector i is the unit vector whose direction is along the positive
x-axis. Vector j is the unit vector whose direction is along the
positive y-axis.
y
1
j
O
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i
1
x
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Representing Vectors in Rectangular Coordinates
Vector v, from (0, 0) to (a, b), is represented as
v = ai + bj.
The real numbers a and b are called the scalar
components of v. Note that a is the horizontal
component of v, and b is the vertical component of v.
The vector sum ai + bj is called a linear combination
of the vectors i and j. The magnitude of v = ai + bj is
given by
2
v  a b
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Text Example
Sketch the vector v = -3i + 4j and find its magnitude.
Solution For the given vector v = -3i + 4j, a = -3 and b = 4. The vector,
shown below, has the origin, (0, 0), for its initial point and (a, b) = (-3, 4) for
its terminal point. We sketch the vector by drawing an arrow from (0, 0) to (3, 4). We determine the magnitude of the vector by using the distance
formula. Thus, the magnitude is
v  a2  b2
2
 (3)  4
2
Terminal point
5
4
 9  16
3
v = -3i + 4j
2
1
 25  5
-5 -4 -3 -2
-1
-1
-2
-3
1
2
3 4
5
Initial point
-4
-5
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Representing Vectors in Rectangular Coordinates
Vector v with initial point P1 = (x1, y1) and terminal point
P2 = (x2, y2) is equal to the position vector
v = (x2 – x1)i + (y2 – y1)j.
•
Adding and Subtracting Vectors in Terms of i and j
If v = a1i + b1j and w = a2i + b2j, then
v + w = (a1 + a2)i + (b1 + b2)j
v – w = (a1 – a2)i + (b1 – b2)j
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Text Example
If v = 5i + 4j and w = 6i – 9j, find:
a. v + w
b. v – w.
Solution
•
v + w = (5i + 4j) + (6i – 9j)
= (5 + 6)i + [4 + (-9)]j
= 11i – 5j
•
v + w = (5i + 4j) – (6i – 9j)
= (5 – 6)i + [4 – (-9)]j
= -i + 13j
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These are the given vectors.
Add the horizontal components. Add the
vertical components.
Simplify.
These are the given vectors.
Subtract the horizontal components.
Subtract the vertical components.
Simplify.
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Scalar Multiplication with a Vector in Terms of i and j
•
•
If v = ai + bj and k is a real number, then the scalar
multiplication of the vector v and the scalar k is
kv = (ka)i + (kb)j.
Example: If v = 2i - 3j, find 5v and -3v
5v  (5 * 2)i  (5 * 3) j
 10i  15 j
 3v  (3 * 2)i  (3 * 3) j
 6i  9 j
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The Zero Vector
The vector whose magnitude is 0 is called the zero vector, 0. The
zero vector is assigned no direction. It can be expressed in terms of I
and j using
•
0 = 0i + 0j.
Properties of Vector Addition
If u, v, and w are vectors, then the following properties are true.
Vector Addition Properties
1. u + v = v + u
2. (u + v) + w = v + (u + w)
3. u + 0 = 0 + u = u
4. u + (-u) = (-u) + u = 0
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Commutative Property
Associative Property
Additive Identity
Additive Inverse
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Properties of Vector Addition and Scalar Multiplication
If u, v, and w are vectors, and c and d are scalars, then the following
properties are true.
Scalar Multiplication Properties
1. (cd)u = c(du)
2. c(u + v) = cu + cv
3. (c + d)u = cu + du
4. 1u = u
5. 0u = 0
6. ||cv|| = |c| ||v||
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Associative Property
Distributive Property
Distributive Property
Multiplicative Identity
Multiplication Property
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Finding the Unit Vector that Has the Same Direction as a Given
Nonzero Vector v
For any nonzero vector v, the vector
v
v
is a unit vector that has the same direction as v. To find this
vector, divide v by its magnitude.
Example
Find a unit vector in the
same direction as v=4i-7j
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v 
4 2  (7) 2
 16  49  65
v
4
7

i
j
v
65
65
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Definition of a Dot Product
If v=a1i+b1j and w = a2i+b2j are vectors, the dot product is defined as
v  w  a1a2  b1b2
The dot product of two vectors is the sum of the products of
their horizontal and vertical components.
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Text Example
If v = 5i – 2j and w = -3i + 4j, find:
a. v · w
b. w · v
c. v · v.
Solution To find each dot product, multiply the two horizontal components,
and then multiply the two vertical components. Finally, add the two products.
a. v · w = 5(-3) + (-2)(4) = -15 – 8 = -23
b. w · v = (-3)(5) + (4)(-2) = -15 – 8 = -23
c. v · v = (5)(5) + (-2)(-2) = 25 + 4 = 29
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Properties of the Dot Product
If u, v, and w, are vectors, and c is a scalar, then
1.
2.
3.
4.
5.
u·v=v·u
u · (v + w) = u · v + u · w
0·v=0
v · v = || v ||2
(cu) · v = c(u · v) = u · (cv)
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Alternative Formula for the Dot Product
•
If v and w are two nonzero vectors and  is the
smallest nonnegative angle between them, then
v · w = ||v|| ||w|| cos.
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Formula for the Angle between Two Vectors
If v and w are two nonzero vectors and  is the
smallest nonnegative angle between v and w, then
vw
1 v  w
cos 
and   cos
v w
v w
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Example
Find the angle  between v=2i-4j and w=3i+2j.
Solution:
  cos1
1
 cos
1
 cos
 cos1
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vw
v w
2 * 3  4 * 2
2 2  ( 4 ) 2 * 3 2  2 2
68
2
1
 cos
20 13
2 65
1
 97.1
65
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The Dot Product and Orthogonal Vectors
Two nonzero vectors v and w are orthogonal if and only if v•w=o.
Because 0•v=0, the zero vector is orthogonal to every vector v.
Example
Are the vectors v=3i-2j and w=3i+2j orthogonal?
v  w  3 * 3  2 * 2
94 5 0
The vectors are not orthogonal.
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The Vector Projection of v Onto w
If v and w are two nonzero vectors, the vector projection of v onto w is
projwv 
Example
vw
2
w
w
If v=3i+4j and w=2i-5j, find the projection of v onto w
Solution:
projw v 
vw
w
2
w
3 * 2  4 * 5
(2i  5 j )
2
2
[2  (5) ]
 14
 28
70

( 2i  5 j ) 
i
j
29
29
29

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The Vector Components of v
Let v and w be two nonzero vectors. Vector v can be expressed
as the sum of two orthogonal vectors v1 and v2, where v1 is
parallel to w and v2 is orthogonal to w.
v1  projwv 
vw
w
2
w, v2  v  v1
Thus, v = v1 + v2. The vectors v1 and v2 are called the vector
components of v. The process of expressing v as v1 and v2 is
called the decomposition of v into v1 and v2.
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Example
Let v=3i+j and w=2i-3j. Decompose v into two vectors, where one is
parallel to w and the other is orthogonal to w.
Solution:
v1 
vw
w
2
w, v2  v  v1
3 * 2  1 * 3
( 2i  3 j )
2
2
2  ( 3)
3
6
9

( 2i  3 j ) 
i
j
13
13
13
6
9
v2  (3 
)i  ( 3 
)j
13
13
33
30

i
j
13
13
v1 
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Definition of Work
•
The work W done by a force F in moving an object
from A to B is
•
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W = F · AB.
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