Download 8.4 notes

Section 8.4 ­ Vectors
A vector in the plane is a line segment with an assigned direction ­­ therefore it has a direction and a magnitude (length)
A
vector is denoted by AB ­ A is the initial point and B is the terminal point
B
The length of the line segment AB is called the magnitude or length of the vector and is denoted by ΙABΙ
Vectors can also be denoted by lower case letters ­­ u
Nov 11­1:12 PM
1
Vector Arithmetic
v
u
v
1. Addition ­­ u + v
2. Subtraction ­­ u ­ v
u
u
­ v
u
u ­ v
3. Scalar Multiplication ­­
u
2u
Nov 11­1:22 PM
2
Vectors in the Coordinate Plane
A vector in the coordinate plane can be described by using its horizontal and vertical components, v = a, b where a is the horizontal component and b is the vertical component
v
b
a
v
b
a
v
b
a
(Remember that a vector represents a magnitude and a direction not a particular arrow. Thus a vector a, b has many different representations depending on its initial point.)
Nov 11­1:40 PM
3
Component Form of a vector ­­ moves the initial point to the origin
If a vector v has initial point P(x1, y1) and terminal point Q(x2, y2)
then v = x2 ­ x1, y2 ­ y1
Examples: Find the component of the vector with initial point P and terminal point Q
P( 3, ­5)
P(­3, ­3)
Q (5, ­8)
Q(6, 0)
Nov 13­6:55 AM
4
Magnitude of a vector ­­ The magnitude or length of a vector
v = a, b is: ΙvΙ =√a2 + b2
b ,find the reference angle
Direction of a vector ­­ tan Θ = a
and place it in the proper quadrant
Example: Find the magnitude and direction of the vector with initial point P(3, ­5) and terminal point Q(­5, ­9)
Nov 13­7:00 AM
5
Algebraic Operations on Vectors
If u = a1, b1 and v = a2, b2 then:
1. u + v = a1 + a2, b1 + b2
2. u ­ v = a1 ­ a2, b1 ­ b2
3. cu = ca1, cb1
Examples: If v = 3, ­4 and w = ­8, ­5 find:
v + w
2w ­ 3v
Nov 13­7:13 AM
6
Vectors in Terms of i and j ­­ The vector v = a, b can be expressed in terms of i and j by v = a, b = ai + bj
(called standard unit vector form)
Example: Write the vector u = ­6, 8 in terms of i and j (standard unit vector form)
Nov 13­10:52 AM
7
Horizontal and Vertical Components of a Vector ­­
If v is a vector with magnitude ΙvΙ and direction θ then
v = a, b = ai + bj where a = ΙvΙ cos θ and b = ΙvΙ sin θ
θ + ΙvΙ j sin θ
Thus we can express v as v = ΙvΙ i cos 2π
Example: A vector has length 10 and direction . Find 3
the horizontal and vertical components and write the vector in terms of i and j.
Nov 16­9:23 AM
8
Modeling Velocity
Velocity ­­ modeled by a vector whose direction is the direction of motion and whose magnitude is the speed.
True velocity ­­ the sum of the vectors affecting the object
True speed ­­ the magnitude of the true velocity
Example: A jet is flying through a wind that is blowing with a speed of 55 mi/h in the direction of N 30o E. The jet has a speed of 765 mi/h relative to the air, and the pilot heads the jet in the direction N 45o E.
a) find the vector w and J that represents the wind and jet respectively
b) find the sum of the vectors
c) the true speed and direction of the jet
Nov 16­1:39 PM
9
Modeling force
Force ­­ the push or pull on an object
Resultant force ­­ sum of the forces acting on an object
Example: Given that forces F1 = ­10, 3 and F2 = 4, ­10
are acting on a point, find the resultant force and magnitude
Nov 16­2:23 PM
10