Post on 20-Jan-2016
6.1 – Vectors in the Plane
What are Vectors?
Vectors are a quantity that have both magnitude (length) and direction, usually represented with an arrow:
• This includes force, velocity, and acceleration
• Component Form:v = <-2,3>
Naming Vectors
A vector can also be written as the letters
of its head and tail with an arrow above:
A – initial pointB – terminal point
Scalars
A quantity with magnitude alone, but no directions, is not
a vector, it’s called a scalar
For example, the quantity “60 miles per hours” is a
regular number, or scalar. The quantity “60 miles per hour
to the northwest” is a vector, because it has both size and
direction
Components
To do computations with vectors, we place them in the plane and find their components.
v
(2,2)
(5,6)
Components
The initial point is the tail, the head is the terminal point. The components are obtained by subtracting coordinates of the initial point from those of the terminal point.
v
(2,2)
(5,6)
Components
The first component of v is 5 -2 = 3. The second is 6 -2 = 4. We write v = <3,4>
v
(2,2)
(5,6)
Magnitude of a Vector
The magnitude (or length) of a vector is shown by two vertical bars on either side of the vector: |a|
OR it can be written with double vertical bars: ||a||
1 1 2 2
2 2
2 1 2 1
2 2
If is represented by the arrow from , to , , then
.
If , , then .
x y x y
v x x y y
a b a b
v
v v
Magnitude of a Vector
Find the magnitude of the vector:
V = <-2,3>
Finding Magnitude of a Vector
Find the magnitude of represented by , where (3, 4) and
(5,2).
PQ P
Q
v
2 2
2 1 2 1
2 2
5 3 2 ( 4)
2 10
x x y y
v
Showing Vectors are Equal
Let u be the vector represented by the directed line segment from R to S, and v the vector represented by the directed line segment from O to P. Prove that u =v.
Addition
To add vectors, simply add their components.
For example, if v = <3,4> and w = <-2,5>,
then v + w = <1,9>.
Multiples of Vectors
Given a real number c, we can multiply a vector by c by multiplying its magnitude by c:
v2v -2v
Notice that multiplying a vector by anegative real number reverses the direction.
Scalar Multiplication
To multiply a vector by a real number, simply multiply each component by that number.
If v = <3,4> and w = <-2,5>, then:
-2v =
4v – 2w =
Vector Operations Example
Let 2, 1 and 5,3 . Find 3 . u v u v
Vector Operations Example
Let 2, 1 and 5,3 . Find 3 . u v u v
3 3 2 , 3 1 = 6, 3
3 = 6, 3 5,3 6 5, 3 3 11,0
u
u v
Unit Vectors
A unit vector is a vector with magnitude (length) of 1.
Given a vector v, we can form a unit vector by multiplying the vector by 1/||v||.
Or you can think of this as v/||v|| (The vector divided by its magnitude)
Finding a Unit Vector
Find a unit vector in the direction of 2, 3 . v
Finding a Unit Vector
Find a unit vector in the direction of 2, 3 . v
222, 3 2 3 13, so
1 2 32, 3 ,
13 13 13
| |
| |
v
v
v
Standard Unit Vectors
A vector such as <3,4> can be written as 3<1,0> + 4<0,1>.
For this reason, these vectors are given special names: i = <1,0> and j = <0,1>.
A vector in component form v = <a,b> can be written ai + bj.
For example, rewrite the vector <-3, 2>
Direction Angles
The precise way to specify the direction of a vector is to state its direction angle (not its slope).
v
Direction Angles
If has direction angle , the components of can be computed
using the formula = | | cos , | | sin .
From the formula above, it follows that the unit vector in the
direction of is cos ,sin .| |
v v
v v v
vv u
v
Finding the components of a Vector
Find the components of the vector with direction angle 120 and
magnitude 8.
v
Finding the components of a Vector
Find the components of the vector with direction angle 120 and
magnitude 8.
v
, 8cos120 ,8sin120
1 3 8 ,8
2 2
4,4 3
So 4 and 4 3.
a b
a b
v
Examples
Find the component form of v, with magnitude 15 and a direction angle of 40 degrees.
Find the component form of vector v with magnitude 6 and direction angle of 115 degrees.
Examples
Find the component form of v, with magnitude 15 and a direction angle of 40 degrees.
<15 cos 40, 15 sin 40> = <11.491, 9.642>
Find the component form of vector v with magnitude 6 and direction angle of 115 degrees.
<6 cos 115, 6 sin 115> = <-2.536, 5.438>
Finding the Direction Angle of a Vector
Find the magnitude and direction angle of 2,3 .u
Finding the Direction Angle of a Vector
2 2|| || 2 3 13
Let be the direction angle of , then
2,3 13 cos , 13sin
2 13 cos
56.3
u
u
u
Find the magnitude and direction angle of 2,3 .u
Finding the direction angle
Find the direction angle for the vector <8, -4>
v
Velocity and Speed
The velocity of a moving object is a vector because velocity has both magnitude and direction.
The magnitude of velocity is speed.
Word Problem
An airplane is flying on a compass heading (bearing) of 170 degrees at 460 mph. A wind is blowing with a bearing of 200 degrees at 80 mph.
a)Find the component form of the velocity of the airplane
b)Find the actual ground speed and direction of the plane.