Unit 2 -Mechanics in Two
Dimensions.
Module 5 –lesson 1
Vectors
Learning objectives
1) Define Vectors in Two Dimensions
2) Use Pythagorean theorem to find the net vector
3) Find the vector components
4) Apply Algebraic Addition of Vectors
Focus Question
How can you add forces in two
dimensions?
New Vocabulary
components
vector resolution
Review Vocabulary
vector: a quantity, such as position, that has
both magnitude and direction.
Vectors in Two Dimensions
• You can add vectors by placing
them tip-to-tail.
• When you move a vector, do not
change its length or direction.
• The resultant of the vector is
drawn from the tail of the first
vector to the tip of the second
vector.
Vectors in Two Dimensions
• If vectors A and B are
perpendicular, use the
Pythagorean theorem to find
the magnitude of the
resultant vector.
Pythagorean Theorem
R2 = A2 + B2
Vectors in Two Dimensions
• If the angle between the
vectors is not 90°, you can
use the law of sines or the
law of cosines.
Law of sines
Law of cosines
R
A
B
=
=
sin q sina sinb
R2 = A2 + B2 - 2AB cos q
Vectors in Two Dimensions
Use with Example Problem 1.
Problem
Find the magnitude of the sum of two
forces, one 20.0 N and the other 7.0 N,
when the angle between them is 30.0°.
Response
SKETCH AND ANALYZE THE PROBLEM
• Draw a vector diagram and add the vectors
graphically.
• List the knowns and unknowns.
Draw
the
Place
Drawvectors
the initial
resultant
tip vectors.
to tail.
vector.
Click
continue.
Click
Clicktoto
tocontinue.
continue.
?
7.0 N
30.0°
20.0 N
θ = 150.0°
KNOWN
A = 20.0 N
B = 7.0 N
θ = 180.0° − 30.0° = 150.0°
UNKNOWN
R=?
θR = ?
Vectors in Two Dimensions
Use with Example Problem 1.
Problem
Find the magnitude of the sum of two
forces, one 20.0 N and the other 7.0 N,
when the angle between them is 30.0°.
KNOWN
A = 20.0 N
B = 7.0 N
θ = 180.0° − 30.0° = 150.0°
UNKNOWN
R=?
θR = ?
SOLVE FOR THE UNKNOWN
• Use the law of cosines.
R 2  A2  B 2  2 AB cos 
Response
 20.0 N  7.0 N
SKETCH AND ANALYZE THE PROBLEM
• Draw a vector diagram and add the vectors
graphically.
• List the knowns and unknowns.
 400 N2  49 N2  280 N2 0.866
2
 220.0 N7.0 Ncos 150.0
R 
?
7.0 N
20.0 N
θ = 150.0°
2
691.49 N2  26.3 N
EVALUATE THE ANSWER
• This answer is consistent with this
problem’s vector diagram, which shows
that the resultant should indeed be
slightly greater in magnitude than the
20.0-N force.
Vector Components
• A vector can be broken into its components, which
are a vector parallel to the x-axis and another
parallel to the y-axis.
• The process of breaking a vector into its
components is sometimes called a vector
resolution.
Vector Components
Algebraic Addition of Vectors
• Two or more vectors (A, B, C, etc.) may be
added by first resolving each vector into its xand y-components.
Algebraic Addition of Vectors
• The x- and y-components are added to form
the x- and y- components of the resultant:
Rx = Ax + Bx + Cx
Ry = Ay + By + Cy
Algebraic Addition of Vectors
• Use the Pythagorean theorem to find the
magnitude of the resultant vector:
R2 = Rx2 + Ry2
Algebraic Addition of Vectors
• Use trigonometry to find the angle of the
resultant vector:
æRy æ
q = tan æ æ
æRx æ
-1
+y
(North)
Algebraic Addition of Vectors
B = 7.3 m
Use with Example Problem 2.
Problem
By
Add the following two vectors via the
component method:
A is 4.0 m south B is 7.3 m northwest
θB = 135°
Response
SKETCH AND ANALYZE THE PROBLEM
• Establish a coordinate system.
• Draw a vector diagram. Include the knowns
and unknowns in your diagram.
• Sketch the x-components and ycomponents of A and B.
Bx
θA = 270°
A = 4.0 m
Ay
+x
(East)
+y
(North)
Algebraic Addition of Vectors
B = 7.3 m
Use with Example Problem 2.
Problem
By
Add the following two vectors via the
component method:
A is 4.0 m south B is 7.3 m northwest
Response
SKETCH AND ANALYZE THE PROBLEM
• Use your vector diagram as needed.
SOLVE FOR THE UNKNOWN
• Find the x-components of A and B and add them
to find the x-component of R.
Ax  4.0 mcos 270  0
 Bx  7.3 mcos 135  5.16 m
Rx  5.16 m
•
Draw and label Rx on the vector diagram.
Bx
+x
(East)
Rx = −5.16 m
A = 4.0 m
Ay
+y
(North)
Algebraic Addition of Vectors
B = 7.3 m
Use with Example Problem 2.
Problem
By
Add the following two vectors via the
component method:
A is 4.0 m south B is 7.3 m northwest
Ry = 1.16 m
Response
SKETCH AND ANALYZE THE PROBLEM
• Use your vector diagram as needed.
SOLVE FOR THE UNKNOWN
• Find the y-components of A and B and add them
to find the y-component of R.
Ay  4.0 msin 270  4.0 m
 By  7.3 msin 135  5.16 m
Ry  1.16 m
•
Draw and label Ry on the vector diagram.
Bx
+x
(East)
Rx = −5.16 m
A = 4.0 m
Ay
+y
(North)
Algebraic Addition of Vectors
B = 7.3 m
Use with Example Problem 2.
Problem
Add the following two vectors via the
component method:
A is 4.0 m south B is 7.3 m northwest
R = 5.3 m
Response
θR = 167°
SKETCH AND ANALYZE THE PROBLEM
• Use your vector diagram as needed.
SOLVE FOR THE UNKNOWN
• R = 5.3 m at 13° north of west
EVALUATE THE ANSWER
• Check the answer graphically.
+x
(East)
A = 4.0 m
Quiz
1. How can vectors be added?
A
by placing them tip-to-tip
B
by placing them tail-to-tail
C
by placing them tip-to-tail
D
by rotating one so it is perpendicular to the
other
CORRECT
Quiz
2. Can the Pythagorean theorem be
used to calculate the length of R?
A
Yes, that method works
for any two vectors.
B
Yes, because R is longer than A and B.
C
No, because the angle between A and B is
not 90°.
D
No, because A and B are not the same length.
CORRECT
Quiz
3. During which process can a vector be
broken into its x- and y-components?
A
axis separation
C
vector resolution
CORRECT
B
vector breakdown
D
vector addition
Quiz
4. Which can be used to find the sum of two vectors when
the vectors are not perpendicular?
A
Pythagorean theorem
C
law of cosines
B
law of sines
D
either B or C
CORRECT
Quiz
5. What is the magnitude of the resultant of two
perpendicular vectors with lengths 3 and 4?
A
B
5
7
CORRECT
C
25
D
625