H.Melikyan/12001 Vectors Dr.Hayk Melikyan Departmen of Mathematics and CS [email protected].
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Transcript of H.Melikyan/12001 Vectors Dr.Hayk Melikyan Departmen of Mathematics and CS [email protected].
H.Melikyan/1200 2
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.
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.
P
Q
Initial point
Terminal point
Directed Line Segments and Geometric Vectors
H.Melikyan/1200 3
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
H.Melikyan/1200 4
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.
Initial point of u
u + vv
u
Resultant vector
Terminal point of v
The Geometric Method for Adding Two Vectors
H.Melikyan/1200 5
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.
v
u-u
-u
v – u
The Geometric Method for the Difference of Two Vectors
H.Melikyan/1200 6
1
1
i
j
Ox
y
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.
H.Melikyan/1200 7
Representing Vectors in Rectangular CoordinatesVector v, from (0, 0) to (a, b), is represented as v = ai + bj.The real numbers a and b are called the scalarcomponents of v. Note that a is the horizontalcomponent of v, and b is the vertical component of v.The vector sum ai + bj is called a linear combinationof the vectors i and j. The magnitude of v = ai + bj isgiven by
v a2 b2
H.Melikyan/1200 8
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
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1-2
-3
-4-5
Initial point
Terminal point
v = -3i + 4j
v a2 b2
( 3)2 42
9 16
25 5
Text Example
H.Melikyan/1200 9
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
H.Melikyan/1200 10
If v = 5i + 4j and w = 6i – 9j, find: a. v + w b. v – w.
Solution • v + w = (5i + 4j) + (6i – 9j) These are the given vectors.
= (5 + 6)i + [4 + (-9)]j Add the horizontal components. Add the vertical components.
= 11i – 5j Simplify.
• v + w = (5i + 4j) – (6i – 9j) These are the given vectors.= (5 – 6)i + [4 – (-9)]j Subtract the horizontal components.
Subtract the vertical components.= -i + 13j Simplify.
Text Example
H.Melikyan/1200 11
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
ji
jiv
ji
jiv
96
)3*3()2*3(3
1510
)3*5()2*5(5
H.Melikyan/1200 12
The Zero VectorThe vector whose magnitude is 0 is called the zero vector, 0. Thezero vector is assigned no direction. It can be expressed in terms of Iand 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 Commutative Property 2. (u + v) + w = v + (u + w) Associative Property 3. u + 0 = 0 + u = u Additive Identity 4. u + (-u) = (-u) + u = 0 Additive Inverse
H.Melikyan/1200 13
Properties of Vector Addition and Scalar MultiplicationIf u, v, and w are vectors, and c and d are scalars, then the followingproperties are true.
Scalar Multiplication Properties 1. (cd)u = c(du) Associative Property 2. c(u + v) = cu + cv Distributive Property 3. (c + d)u = cu + du Distributive Property 4. 1u = u Multiplicative Identity 5. 0u = 0 Multiplication Property 6. ||cv|| = |c| ||v||
H.Melikyan/1200 14
Finding the Unit Vector that Has the Same Direction as a Given Nonzero Vector v
For any nonzero vector v, the vector
is a unit vector that has the same direction as v. To find this
vector, divide v by its magnitude.
v
v
Example
Find a unit vector in the same direction as v=4i-7j
jiv
v
v
65
7
65
4
654916
)7(4 22
H.Melikyan/1200 15
Definition of a Dot Product
If v=a1i+b1j and w = a2i+b2j are vectors, the dot product is defined as
2121 bbaawv The dot product of two vectors is the sum of the products of their horizontal and vertical components.
H.Melikyan/1200 16
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
Text Example
H.Melikyan/1200 17
Properties of the Dot Product
If u, v, and w, are vectors, and c is a scalar, then 1. u · v = v · u 2. u · (v + w) = u · v + u · w 3. 0 · v = 0 4. v · v = || v ||2
5. (cu) · v = c(u · v) = u · (cv)
H.Melikyan/1200 18
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.
H.Melikyan/1200 19
Formula for the Angle between Two Vectors
wv
wvand
wv
wv
1coscos
If v and w are two nonzero vectors and is thesmallest nonnegative angle between v and w,
then
H.Melikyan/1200 20
Example
Find the angle between v=2i-4j and w=3i+2j.
Solution:
1.9765
1cos
652
2cos
1320
86cos
23*)4(2
2*43*2cos
cos
1
11
2222
1
1
wv
wv
H.Melikyan/1200 21
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?
0549
2*23*3
wv
The vectors are not orthogonal.
H.Melikyan/1200 22
The Vector Projection of v Onto w
If v and w are two nonzero vectors, the vector projection of v onto w is
ww
wvvprojw 2
If v=3i+4j and w=2i-5j, find the projection of v onto wSolution:
jiji
ji
ww
wvvprojw
29
70
29
28)52(
29
14
)52(])5(2[
5*42*322
2
Example
H.Melikyan/1200 23
The Vector Components of v
Let v and w be two nonzero vectors. Vector v can be expressedas the sum of two orthogonal vectors v1 and v2, where v1 is
parallel to w and v2 is orthogonal to w.
1221 , vvvww
wvvprojv w
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.
H.Melikyan/1200 24
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:
ji
jiv
jiji
jiv
vvvww
wvv
13
30
13
33
)13
93()
13
63(
13
9
13
6)32(
13
3
)32()3(2
3*12*3
,
2
221
1221
H.Melikyan/1200 25
Definition of Work
• The work W done by a force F in moving an object from A to B is
• W = F · AB.