Transcript No Slide Title

Algebraic Addition
of Vectors
Now you shall cast aside your
rulers and protractors, and use
your head and calculators to add
vectors…
Adding Vectors in One
Dimension
4N
2N
+
4N
=
2N
+
=
6N
2N
This is so easy!!!
When do I need a calculator??
Adding Vectors In Two Dimensions
If you’re really lucky, the vectors might be
perpendicular to each other:
A B C
2
B
A

C
2
Where A and B are the
magnitudes of A and B,
and C is the magnitude
of C.
B
  tan  
A
-1
Always use the sides
given, not calculated
2
What If You’re Not So Lucky?
Well, the process is a little more complicated.
First we need to talk about components of
vectors.
Any vector may be considered the vector sum
of an infinite number of other vector
combinations:
Components of Vectors
Any vector may be expressed as the sum of a
vector parallel to the horizontal axis (Ax), and
a vector parallel to the vertical axis (Ay):
A
Ay
Ax
 

A  Ax  Ay
A  A A
2
2
x
2
y
How Can We Find These?
Since Ax is perpendicular
to Ay the following is true:
Ax
cos θ 
A
sinθ 
Ay
A
A cosθ  A x
A sinθ  A y
A
Ay

Ax
What Good are These Components?
By
Cy
C
A
Ay
Ax
Cx
B
Bx
Steps for Adding Vectors…
… That aren’t parallel or perpendicular to each
other:
1) Break all vectors into their horizontal and vertical
components.
2) Add the horizontal components together to get the
horizontal component of the resultant.
3) Add the vertical components together to get the
vertical component of the resultant.
4) Add the components of the resultant using the
Pythagorean Theorem.
5) Use the components of the resultant and a
trigonometric function to get the direction of the
resultant.
For Example:
Let’s Add Vectors A and B:
A

B

A = 13 N @ 22.6
B = 5 N @ 126.9
(1) Break all vectors into their horizontal
and vertical components.
A
Ay
Ax
A = 13 N @ 22.6
Ax= 13 N cos 22.6  = 12 N
Ay = 13 N sin 22.6 = 5 N
B = 5 N @ 126.9
Bx= 5 N cos 126.9  = 3 N
By = 5 N sin 126.9 = 4 N
By
B
Bx
(2) And (3) Add the Components
Ax= 13 N cos 22.6  = 12 N
Bx= 5 N cos 126.9  = 3 N
Ax + Bx= Cx = 9 N
Ay = 13 N sin 22.6 = 5 N
By = 5 N sin 126.9 = 4 N
Ay + By= Cy = 9 N
(4) Add the Components for the Resultant
C2 = Cx2 + Cy2
C2 = (9 N)2 + (9 N)2
C
C = 12.7 N
Cy = 9 N
Cx = 9 N
The components will ALWAYS be
perpendicular to each other, so
we’ll always use the Pythagorean
Theorem
(5) Find the Angle
 9N
  tan 

 9N
-1
C
 9N
  tan 

 9N
 9N

 9N
Cy = 9 N
  tan-1
-1
 45

Cx = 9 N
You know the legs of the
right triangle, as well as
the hypotenuse, so you can
use any trig function…
Find the Vector Sum of A + B
C= 12.7 N @ 45
C
B
A