Vectors in Two
Dimensions
Read Chapter 1.6-1.9
Scalars and VectorsAll measurements are considered to be
quantities. In physics, there are 2 types of quantities – SCALARS AND VECTORS.
Scalar quantities have only magnitude.
Vector quantities have magnitude and direction.
time mass
temperature
displacement
velocity
acceleration
Gravitational Field
Magnetic Field
Force
Vectors are used to describe motion and solve problems concerning motion.
For this reason, it is critical that you have an understanding of how to
represent vectors add vectorssubtract vectorsmanipulate vector
quantities.in 2 and 3 dimensions.
Vectors
tail
tip
Magnitude represented by the length of the vector
8 units5 units2 units
Vectors
Direction represented by the direction of the arrow
x
y
003008001200600 from -x2250
450
from-y
3000
-600
Adding VectorsWe know how to add vectors in 1-dimension.
Example: displacement. If someone walks 4 mi east (Dx1) and then 7 mi west (Dx2) the total displacement (Dx1 + Dx2) is 3 mi west.1. Adding vectors GRAPHICALLY – place them TAIL TO TIP
2. Adding vectors MATHEMATICALLY – In 1- dimension, assign direction + or – and add algebraically
Dx1 + Dx2 = +4 mi + (-7 mi) = -3 mi
east
+west
-
Dx1 = 4 mi
Dx2 = 7 mi
Dx1 + Dx2 = 3 mi
Adding VectorsWhat about if the vectors are in
different directions in 2-D?How do we describe the direction?How do we add/subtract the vectors?
v 1 =
5
v2 = 350o above +x
35o below +x
Adding Vectors Graphically“tail to tip”To add vectors using the
tail to tip method
1. Draw the first vector (7u, 50o N of E) beginning at the origin.
2. Draw the second vector (3u, 35o S of E) with its tail at the tip of the first vector.
3. Draw the Resultant vector (the answer) from the tail of the first vector to the tip of the last.
North
South
EastWest
v1
v2
v = v 1 + v 2
Adding Vectors Grahically
“parallelogram method”To add vectors with the parallelogram method
1.Draw the first vector to scale beginning at the origin.
2.Draw the second vector, to scale, with its tail also at the origin.
3.Starting at the tip of one vector, draw a dotted line parallel to the other vector. Repeat, starting from the tip of the second vector.
4.Draw the Resultant vector (the answer) from the origin to the intersection of the dotted lines.
North
South
EastWest
v1
v2
v = v 1 + v 2
Adding Vectors MathematicallyWhat if the vectors are in different directions?
For example, what if I walk 5 steps north and then 4 steps east. What is my total displacement , Dx, for the trip?
OR what is the vector sum of the Dx1 and Dx2?
Dx2 =4 steps east
Dx1=5 steps north Dx = Dx1+Dx2 = ?
Adding Vectors Mathematically4 steps east
5 steps north Dx = ?
Use Pythagorean Theorem to find the magnitude of Dx
steps 4.64145 222
xx
Use right triangle trig to find the direction of Dx
north ofeast 7.38)8.0(tan
8.05
4tan
0
1
q
RIGHT TRIANGLE TRIGONOMETRY
A
OH
q
A – side adjacent to angle qO – side opposite to angle qH – hypotenuse of triangle
sinq = OH
cosq = AH
tanq = sinqcosq =
OA
Pythagoreon Theorem 222 AOH
SOHCAHTOA
A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip?
West
tanq = oppadj
150m240m=
tanq = 0.625q= tan-1(1.3) = 32o south of east
MAGNITUDE R2 = A2 +B2
= 2402 + 1502
R = 283 mDIRECTION
North
South
EastA = 240 m
B=150m
qR=283 m
Notice that A and B are perpendicular components of R. They are the amount of R in each direction
A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip?
West
North
South
East240 m
150mq283 m
The net displacement is 283 m in the direction of 32o S of E.
The average velocity:
v = DxDt
283 m0.5 hr=
= 566 m/hr in the direction 32o S of E.
EXAMPLE: What is the Resultant of adding 2 vectors, A and B, if A = 8 units south and B = 4.5 u west?
North
South
EastWest
4.5u west
8u southq
tanq = oppadj
4.5 u8 u=
tanq = 0.56
q-1 = tan-1(0.56) = 29o 9.2 u
The Resultant is 9.2 units in the direction of 29o south of west
MAGNITUDE R2 = A2 +B2
= 82 + 4.52
R = 9.2 uDIRECTION
R
Notice that A and B are perpendicular components of R. They are the amount of R in each direction
30o
Ex. Vector A has magnitude 8.0 m at an angle of 30 degrees below the x-axis. What are the x- and y-components of A?
A=8Ay
Ax
sin30 = Ay
AAy = 8sin30 = 8(0.5) = - 4m
cos30 = Ax
AAx = 8cos30 = 8(0.866) = +6.9
Perpendicular Components of a Vector Any vector can be resolved into perpendicular components. Use right triangle trig – x and y components always make a right triangle with the vector .
Ax = Acos30
Ay = Asin30
You must assign the correct direction
What are the x- and y- components of the vector A, shown below?
A = 8 m
30o x
y
Ax
Ay Ax = Acosq
= 8cos(30)
= 8(0.866)
= -6.9 m Ay = Asinq
= 8sin(30)
= 8(0.5)
= +4 m
x- and y- Components of a Vector
Ax is the amount of A in the x-direction Ay is the amount
of A in the y-direction
North
South
East
West ?q
A butterfly moves with a speed of 12 m/s. The x-component of its velocity is 8.00 m/s. The angle between the direction of its motion and the x-axis must be what?
v = 12 m/s
vy
vx = 8cosq = vx
v
q = cos-10.667 = 48o
812=
Adding Vectors by ComponentsWhat about adding 2 vectors, A and B, that are NOT
perpendicular or parallel?
y
x
Ry
60o
A
B By
Bx
R
Rx
Any vector can be described as the sum of perpendicular components.
Components in the same direction are added as 1-D vectors to find the components of the resultant vector.
Rx = Ax +
Bx
= 0 + Bx
Ry = Ay +
By
(Ax+Ay)+(Bx+By)=(Rx+Ry)
A + B =
R
Adding x- and y- Components of a Vector
Ry
60o
A
B By
Bx
R
Rx
R2 = Rx2 + Ry
2
x Y
A Ax Ay
B Bx By
R Rx Ry
A + B =
R Determine the perpendicular components of each vector. Make a table to add up x and y components separately:
q
Magnitude of R:
x
y
x
y
R
R
R
R 1tantan Direction of R:
(Ax+Ay)+(Bx+By)=(Rx+Ry)
y
x
ADDING VECTORS GRAPHICALLYYou can add as many vectors as
you wantNorth
South
West
B
A
R
C
D
East
A + B + C + D =
R Graphically, the vectors are added “tail to tip”
and the order doesn’t matter
C + A + D+ B = R
A
B C
D
A + B + C + D =
R To Add Vectors by Components:
1.Place each vector at origin. Find the x- and y- components of each vector in the sum, and list them in a table. Make sure to include the direction of each component by hand.
2.Add all the x-components to find Rx.
3.Add all the y-components to find Ry.
4.Use Pythagorean Theorem and trigonometry to find the magnitude and direction of R.
Mathematically ADDING VECTORS (Ax+Ay)+(Bx+By)+(Cx+Cy)+
(Dx+Dy)=(Rx+Ry) x Y
A -Ax -Ay
B+B
x-By
C+C
x
+Cy
D -Dx+D
y
R
Rx Ry
R2 = Rx2 + Ry
2Magnitude of R:
x
y
R
R1tan Direction of R:
EXAMPLE (Adding Vectors by Components)
Determine the resultant of the following 3 displacements:
A. 24m, 30º north of eastB. 28m, 37º east of northC. 20m, 50º west of south
x (m) y (m)
A 20.8 12.0
B 16.9 22.4
C -15.3 -12.9
Ʃ 22.4 21.5
North
South
West
A B
C
East
30o
37o
50o
Ay
Ax
By
Bx
Cy
Cx
EXAMPLE
x (m) y (m)
A 20.8 12.0
B 16.9 22.4
C -15.3 -12.9
Ʃ R 22.4 21.5
North
South
West
R
East
m315.214.22 22
22
222
yx
yx
RRR
RRR
Ry
Rx
Magnitude Direction
01 4496.0tan4.22
5.21tan
x
y
R
R
q
R = 31 m, 440 N of E (from the x-axis)
DO NOWAn airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement?
North
South
West A
B C
East
45o53o By Cy
Cx Bx
x (km)
y (km)
A 620 0
B 311 -311
C -331 -439
DR 600 -750
DR
EXAMPLE (cont.)North
South
West
DR
East
q
x (km)
y (km)
A 620 0
B 311 -311
C -331 -439
DR 600 -750
m960750600 22
222
kR
RRR yx
Magnitude Direction
01 51)25.1(tan600
750tan
x
y
R
R
DR = 960 km, -510 from the x-axis (510 S of E)
EXAMPLEAn airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement?
North
South
West
DR
East
q
960 km at 510 South of East
Subtracting VectorsIn order to subtract a vector, we add the negative of that vector.
The negative of a vector is defined as a vector in the OPPOSITE direction (with each component the negative of the original)v1 v2- v1 -v2+=
v1
-v2
= v1
v2
-
A=8 B=10
60o
A + B = S
A – B = D
AB
S A
B D
D -B
A
Graphical Representation
Tail to tip Tail to tip(A and –B)
Tail to tail(A and B)
-B60o
A=8 B=10
60o
A + B = S
A + (–B) = D
Mathematical Representation
x y
A + 8 0 B + 5 - 8.7
S 13 -8.7
x y
A + 8 0
-B - 5+
8.7
D 3 +8.7
S
Sx
Syq D
Dx
Dyq
-B
60o
Scalar MultiplicationMultiplication of a vector by a positive scalar changes
the magnitude of the vector, but leaves its direction unchanged. The scalar changes the size of the vector. The scalar "scales" the vector.
Multiplication of a vector by a negative scalar changes the magnitude of the vector, and makes the direction opposite.
Example:
3A = 3 x = =
If the scalar is negative, it changes the direction of the vector.
-3A = -3 x = =
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