Transcript Sullivan Algebra and Trigonometry: Section 10.4

Sullivan Algebra and
Trigonometry: Section 10.4
Objectives of this Section
• Graph Vectors
• Find a Position Vector
• Add and Subtract Vectors
• Find a Scalar Product and Magnitude of a Vector
• Find a Unit Vector
• Find a Vector from its Direction and Magnitude
• Work With Objects in Static Equilibrium
A vector is a quantity that has both
magnitude and direction.
Vectors in the plane can be
represented by arrows.
The length of the arrow represents the
magnitude of the vector.
The arrowhead indicates the direction
of the vector.
Q
Terminal
Point
Initial
Point
P


Directed line segment PQ

The magnitude of the directed line segment PQ
is the distance for the point P to the point Q.

The direction of PQ is from P to Q.
If a vector v has the same magnitude and
the same direction as the directed line
segment PQ, then we write

v  PQ
The vector v whose magnitude is 0 is called
the zero vector, 0.
Two vectors v and w are equal, written
v  w if they have the same magnitude
and direction.
v
w
vw
Vector Addition
vw
Initial
point of v
v
Terminal
point of w
w
Vector addition is commutative.
vwwv
Vector addition is associative.
u  v  w   u  v   w
v00v  v
v   v   0
Multiplying Vectors by Numbers
If  is a scalar (a real number) and v is a vector,
the scalar product v is defined as
1. If  > 0, the product v is the vector
whose magnitude is  times the magnitude
of v and whose direction is the same as v.
2. If  < 0, the product v is the vector
whose magnitude is  times the magnitude
of v and whose direction is opposite that of v.
3. If  = 0 or if v  0, then v  0.
Properties of Scalar Products
0v  0
1v  v
 1v  v
v  w  v  w
  v  v  v
v  v
Use the vectors illustrated below to graph
each expression.
w
v
u
v
w
vw
2v
v
w
w
2 v and - w
w
2v
2v  w
If v is a vector, we use the symbol v to
represent the magnitude of v.
If v is a vector and if  is a scalar, then
(a) v  0
(b) v  0 if and only if v  0
(c)  v  v
(d) v   v
A vector u for which u  1 is called a
unit vector.
Let i denote a unit vector whose direction
is along the positive x-axis; let j denote a
unit vector whose direction is along the
positive y-axis. If v is a vector with initial
point at the origin O and terminal point at
P = (a, b), then
v  ai  bj
P = (a, b)
b
bj
ai
a
The scalars a and b are called
components of the vector v = ai + bj.
Suppose that v is a vector with initial point
P1  x1 , y1 , not necessarily the origin, and

terminal point P2  x2 , y 2 . If v  P1 P2 , then
v is equal to the position vector
v  x2  x1 i   y2  y1 j
Find the position v ector of the vector

v  P1 P2 if P1   2,1and P2  3,4 .
v  x2  x1 i   y2  y1 j
v  3  ( 2) i  4  1j
v  5i  3 j
P2  3,4 
P1   2,1
O
5,3
Equality of Vectors
Two vectors v and w are equal if and
only if their corresponding
components are equal. That is,
If v  a1i  b1j and w  a2i + b2 j,
then v  w if and only if a1  a2 and b1  b2 .
Let v  a1i  b1j and w  a2i  b2 j be two
vectors, and let  be a scalar. Then,
v  w  a1  a2 i  b1  b2 j
v  w  a1  a2 i  b1  b2 j
v  a1 i  b1 j
v 
2
a1
2
 b1
If v  3i  2 j and w  4i  j, find
(a) v  w
(b) v  w
(a) v  w  3i  2 j   4i  j
 3  4i  2  1j
  i  3j
(b) v  w  3i  2 j   4i  j
 3   4 i  2  1j
 7i  j
If v  3i  2 j and w  4i  j, find
(a) 2 v  3w
(b) v
(a) 2 v  3w  23i  2 j  3 4i  j
 6i  4 j  12i  3j
  6i  7 j
(b) v  3i  2 j  3  2
2
 13
2
Unit Vector in Direction of v
For any nonzero vector v, the vector
v
u
v
is a unit vector that has the same
direction as v.
Find a unit vector in the same
direction as v = 3i - 5j.
v  3  ( 5)  9  25  34
2
2
3
5
v 3i  5j

i
j
u 
34
34
v
34
3 34 5 34

i
j
34
34