1.2 Vectors in Two Dimensions
Defining Vector Components
Any vector can be resolved into endless number of components vectors.
D2
D1
DR
D4
D3
DR
The ability to add components vectors is key to solving all sorts of
Physics problems.
p. 11
1.2 Vectors in Two Dimensions
Trigonometric Rations Used in Vector Problems:
A
hypotenuse
Opposite side
Sin ѳ =
opposite side
ѳ
B
adjacent side
hypotenuse
o
=
h
adjacent side
C
Trigonometric ratios can help you solve vector
problems. The rules for adding velocity vectors
are the same as those for displacement and
force vectors.
Cos ѳ =
hypotenuse
a
=
h
Opposite side
Tan ѳ =
Adjacent side
p. 12
o
=
a
1.2 Vectors in Two Dimensions
Resolving Vectors into Vertical and Horizontal Components
Vertical
Component
Ѳ = 53o
FR = 60.0 N
A force of 60.0 N is applied downwards at
an angle of 53o below the horizontal.
This vector can be show as the addition of
two perpendicular component vectors.
Horizontal
Component
Vertical Component is directed downwards into the ground:
Horizontal Component is directed horizontally to the ground (may be used to move the
object along the ground)
p. 13
1.2 Vectors in Two Dimensions
Method 1: Solving by Scale Diagram
Length = 3.6 cm
Ѳ = 53o
FR = 60.0 N
A scale is used (1.0 cm = 10.0 N) is
drawn over the vectors.
By using this scale the vertical
component can be determined to be
36.0 N and the horizontal component
is 48.0 N.
Fy=
36.0 N
Fx = 48.0 N
Length = 4.8 cm
p. 12
1.2 Vectors in Two Dimensions
Method 2: Resolve into Components
Cos Ѳ =
Cos 53o =
adj
Hyp
FR = 60.0 N
Fy
FR
Fy = cos 53o x 60
Fy = 36 N
Ѳ=
53o
Sin Ѳ =
Sin 53o =
Opp
Hyp
Fx
By using the correct
trigonometric functions
both the vertical and
horizontal components can
be determined.
FR
Fx = sin 53o x 60
Fx = 48 N
p. 13 - 14
1.2 Vectors in Two Dimensions
Method 2: Resolve into Components (con’t)
Fy = 36 N
Ѳ=
53o
FR = 60.0 N
Fx = 48 N
To check to see if you have the right answer use Pythagorean theorem as follows:
FR2 = Fx2 + Fy2
FR2 = 482 + 362
FR = 60 N
p. 13 - 14
1.2 Vectors in Two Dimensions
More than One Vector: Using a Vector Diagram
Two strings support an object.
String #1
30o
String #2
To determine tension in each string
a force triangle is made from the
three forces.
Force of gravity Fg = 36.0 N
Fg = 36 N
30o
Tension #1
Tension #2
p. 14
1.2 Vectors in Two Dimensions
More than One Vector: Using a Vector Diagram
Fg = 36 N
30o
F1 = Tension #1
Cos 30o =
F1 =
Fg
F1
36.0
Sin 30o
F1 = 41.6 N
F2 = Tension #2
Tan 30o =
F2 =
Fg
F2
36.0
Each tension can be found by using the correct
trigonometric function.
Tan 30o
F2 = 20.8 N
p. 15
1.2 Vectors in Two Dimensions
A Velocity Vector Problem – Vectors in Action
Boats crossing rivers and Planes travelling against wind are all ideal vector problems.
head wind
tail wind
vw
vw
vr
vb
vR
vR
vr
ѳ
vp
vb
vp
vR
2
2
vR = v r + vb
2
vR = vp + vw
vR = vp - vw
p. 17 - 20
1.2 Vectors in Two Dimensions
Key Questions
In this section, you should understand how to solve the following key questions.
Page 16 – 17 – Practice Problem 1.2.1 #1 - 3
Page 20 – Practice Problem 1.2.2 #1 – 3
P. 21 – 22 1.2 Review Questions # 1 - 9