Vectors - 2 Dimension
Introduction
Vectors can be added if direction is taken into account.  The
example to the left shows the graphical addition of the red, blue, green, and
purple vectors.
The end of 1 vector is the start of the next vector.
After all 4 vectors have been added; the resultant vector is the vector
starting from the beginning of the first vector, red, to the end of the last
vector, purple.
In this drawing, the resultant vector is the black vector.
The graphical method gives a good view of the problem; however, it is not
the most accurate method.
There are two trig methods that can be used to add vectors.
| 1. |
| a. |
Draw the first two vectors using the
graphical method and complete the parallelogram.
|
| b. |
Draw the resultant vector and use trig
to calculate the resultant vector.
|
| c. |
Draw the resultant vector and the
third vector using the graphical method.
|
| d. |
Draw the resultant vector of step c
and use trig to calculate the resultant vector.
|
| e. |
Continue until you have added all of
the vectors. The last resultant vector is the total resultant
vector.
|
| f. |
This method is best used when there
are only two vectors.
|
|
| 2. |
| a. |
Pick two directions that are at right
angles to each other. Many times we will use North-South vs
East-West or horizontal vs vertical.
|
| b. |
Break each vector into two
components. For each direction assign a positive and negative
direction. If North is positive, then South is
negative.
|
| c. |
Add all of the components in the same
direction.
|
| d. |
Calculate the resultant of the two
total components.
|
| e. |
This method is quicker when there are
multiple vectors being added. It can be used with only two
vectors. It has several advantages when doing projectile
problems. |
|
|
The red vector is 12.0 cm and the blue
vector is 10.0 cm. Calculate the resultant vector.
|
Work
this problem using the 1st method.
|
|
Adjacent angles in the parallelogram
have a total of 180o. |
Complete
the parallelogram.
Calculate the angle opposite the resultant. |
| c2 = a2
+ b2 - 2ab Cos C
c = (a2 + b2 - 2ab Cos
C) 1/2
c = ( (12.0 cm)2 + (10.0 cm)2
- 12.0 cm x 10.0 cm x Cos 102o ) 1/2
Resultant = 16.4 cm
|
Use
the Law of Cosines to find the magnitude of the resultant. |
| Clockwise angle from the red
vector.
Blue vector is opposite this angle.
Sin B Sin
A
b Sin A
------- = ---------- => B = Sin -1 (
------------ )
b
a
a
10.0 cm x Sin 102o
Clockwise angle = Sin -1 ( --------------------------- ) = 36.6o
16.4 cm
The resultant vector is 16.4 cm 36.6o clockwise from the red
vector.
Since the clockwise and counter clockwise angles add up to 78o
,
the counter clockwise angle is 41.4o .
The resultant vector is 16.4 cm 41.4o counter clockwise
from the blue vector. |
Use
the Law of Sines to calculate the angle clockwise angle from the red
vector or the counter clockwise angle from the blue vector. |
Calculate the resultant vector if you add 13.7 cm at 85.0o,
9.32 cm at 125.0o, and 18.4 cm at 235.0o.
 |
|
Use
the Compass to determine the quadrant for each vector.
|
 |
North = 13.7 x Cos 85.0o
= 1.19 cm
East = 13.7 cm x Sin 85.0o
= 13.6 cm
South = 9.32 cm x Cos 55.0o
= 5.35 cm
East = 9.32 cm x Sin 55.0o
= 7.63 cm
South = 18.4 cm x Cos 45.0o
= 13.0 cm
West = 18.4 cm x Sin 45.0o
= 13.0 cm
|
Draw each vector completing the rectangle for
each vector.
(With practice, you can do the work by
visualizing the drawings.)
Break
each vector into components. The components are the sides of the
rectangle.
opposite
Sin = ----------------
hypotenuse
opposite = hypotenuse x Sin
adjacent
Cos = -----------------
hypotenuse
adjacent = hypotenuse x Cos
|
| South positive and North
negative
East positive and West negative
North-South = - 1.19 cm + 5.35 cm + 13.0 cm = + 17.2 cm (South)
East-West = 13.6 cm + 7.63 cm - 13.0 cm = + 8.2 cm (East)
|
For
each dimension, assign a positive and negative.
Add the vectors in the same dimension.
|
Resultant is
19.1 cm at 154.2o |
Resultant =
((17.2 cm)2 + (8.2 cm)2)
1/2
Resultant = 19.1 cm
Angle between East and Vector
17.2 cm
Angle = Sin -1 ( ------------- )
19.1 cm
Angle = 64.2o
Heading = 90.0o + 64.2o
Heading = 154.2o
|
Draw
the two components and solve for the resultant vector using
hypotenuse = (a2 + b2)
1/2
Solve for one of the two angles and convert
into a heading.
opposite
Angle = Sin -1 ( ----------------- )
hypotenuse
|
Table of Contents
Vector Problems
|
Calculate the resultant vector for each of
the following.
|
| 1. |
25.3 cm and 18.6 cm separated by 123.0o.
|
| 2. |
8.64 cm and 9.32 cm separated by 38.4o.
|
| 3. |
5.93 cm at a heading of 75.0o, 3.17 cm at a heading of 125.0o,
8.68 cm at a
heading of 218.0o, and 4.33 cm at a heading of 313.0o.
|
| 4. |
6.89 cm at a heading of 115.0o, 5.11 cm at a heading of 31.0o, 4.22 cm at a
heading of 314.0o, and 8.14 cm at a heading of 193.0o. |
Table of Contents
Vector Answers
|
Calculate the resultant vector for each of
the following.
|
| 1. |
25.3 cm and 18.6 cm separated by 123.0o.
|
|
First
draw the vectors. |
|
Complete the parallelogram.
Angle = 180o - 123.0o
= 57.0o
180o = Sum of adjacent
angles
in a parallelogram
Use
the Law of Cosines to find the magnitude of the resultant.
c2 = a2
+ b2 - 2ab Cos C
=> c = (a2 + b2 - 2ab Cos
C) 1/2
resultant = ((25.3 cm)2 +
(18.6 cm)2 - 2 x 25.3 cm x 18.6 cm x Cos 57o)
1/2
resultant = 21.8 cm
|
|
|
Use
the Law of Sines to calculate the angle clockwise angle from the red
vector or the counter clockwise angle from the blue vector.
Clockwise angle from the red
vector.
Blue vector is opposite this angle.
Sin B Sin
A
b Sin A
------- = ---------- => B = Sin -1 (
------------ )
b
a
a
18.6 cm x Sin 57o
Clockwise angle = Sin -1 ( --------------------------- ) = 45.7o
21.8 cm
The resultant vector is 21.8 cm 45.7o clockwise from the red
vector.
Since the clockwise and counter clockwise angles add up to
123.0o
,
the counter clockwise angle is 77.3o .
The resultant vector is 21.8 cm 77.3o counter clockwise
from the blue vector. |
|
|
|
| 2. |
8.64 cm and 9.32 cm separated by 38.4o.
|
 |
First
draw the vectors. |
|
Complete the parallelogram.
Angle = 180o - 38.4o
= 141.6o
180o = Sum of adjacent
angles
in a parallelogram
Use
the Law of Cosines to find the magnitude of the resultant.
c2 = a2
+ b2 - 2ab Cos C
=> c = (a2 + b2 - 2ab Cos
C) 1/2
resultant = ((8.64 cm)2 + (9.32 cm)2 - 2 x
8.64 cm x 9.32 cm x Cos 141.6o)
1/2
resultant = 17.0 cm
|
|
Use
the Law of Sines to calculate the angle clockwise angle from the red
vector or the counter clockwise angle from the blue vector.
Clockwise angle from the red
vector.
Blue vector is opposite this angle.
Sin B Sin
A
b Sin A
------- = ---------- => B = Sin -1 (
------------ )
b
a
a
9.32 cm x Sin 141.6o
Clockwise angle = Sin -1 ( ----------------------------- ) =
19.9o
17.0 cm
The resultant vector is 17.0 cm 19.9o clockwise from the red
vector.
Since the clockwise and counter clockwise angles add up to
38.4o
,
the counter clockwise angle is 17.4o .
The resultant vector is 17.0 cm 17.4o counter clockwise
from the blue vector. |
|
|
|
| 3. |
5.93 cm at a heading of 75.0o, 3.17 cm at a heading of 125.0o,
8.68 cm at a
heading of 218.0o, and 4.33 cm at a heading of 313.0o.
|
Use
the Compass to determine the quadrant for each vector.
|
 |
North
= 5.93 cm x Cos 75.0o
= 1.53
cm
East = 5.93 cm x Sin 75o
= 5.73 cm
South = 3.17 cm x Cos 55.0o
= 1.82
cm
East = 3.17 cm x Sin 55.0o
= 2.60 cm
South = 8.68 cm x Cos 38.0o
= 6.84 cm
West = 8.68 cm x Sin 38.0o
= 5.34 cm
North = 4.33 cm x Cos 47.0o
= 2.95
cm
West = 4.33 cm x Sin 47.0o
= 3.17 cm
|
Draw each vector completing the rectangle for
each vector.
(With practice, you can do the work by
visualizing the drawings.)
Break
each vector into components. The components are the sides of the
rectangle.
opposite
Sin = ----------------
hypotenuse
opposite = hypotenuse x Sin
adjacent
Cos = -----------------
hypotenuse
adjacent = hypotenuse x Cos
|
| South positive and North
negative
East positive and West negative
North-South = - 1.53 cm + 1.82 cm + 6.84 cm - 2.95= + 4.18 cm (South)
East-West = 5.73 cm + 2.60 cm - 5.34 cm - 3.17
cm = - 0.18 cm (West)
|
For
each dimension, assign a positive and negative.
Add the vectors in the same dimension.
|
 |
Resultant =
((4.18 cm)2 + (0.18 cm)2)
1/2
Resultant = 4.18 cm
Angle between North and Vector
0.18 cm
Angle = Sin -1 ( ------------- )
4.18 cm
Angle = 2.5o
Heading = 360.0o - 2.5o
Heading = 357.5o
Resultant = 4.18 cm at 2.5o
|
Draw
the two components and solve for the resultant vector using
hypotenuse = (a2 + b2)
1/2
Solve for one of the two angles and convert
into a heading.
opposite
Angle = Sin -1 ( ----------------- )
hypotenuse
|
|
|
|
| 4. |
6.89 cm at a heading of 115.0o, 5.11 cm at a heading of 31.0o, 4.22 cm at a
heading of 314.0o, and 8.14 cm at a heading of 193.0o.
|
Use
the Compass to determine the quadrant for each vector.
|
 |
South
= 6.89 cm x Sin 25.0o
= 2.91
cm
East = 6.89 cm x Cos 25.0o
= 6.24 cm
North = 5.11 cm x Cos 31.0o
= 4.38
cm
East = 5.11 cm x Sin 31.0o
= 2.63 cm
South = 4.22 cm x Sin 77.0o
= 4.11
cm
West = 4.22 cm x Cost 77.0o
= 0.95 cm
North = 8.19 cm x Sin 49.0o
= 6.18
cm
West = 8.19 cm x Cos 49.0o
= 5.37 cm
|
Draw each vector completing the rectangle for
each vector.
(With practice, you can do the work by
visualizing the drawings.)
Break
each vector into components. The components are the sides of the
rectangle.
opposite
Sin = ----------------
hypotenuse
opposite = hypotenuse x Sin
adjacent
Cos = -----------------
hypotenuse
adjacent = hypotenuse x Cos
|
| North
positive and South negative
East positive and West negative
North-South = - 2.91 cm + 4.38 cm - 4.11 cm + 6.18 cm
= + 3.54 cm (North)
East-West = 6.24 cm + 2.63 cm - 0.95 cm - 5.37
cm = + 2.55 cm (East)
|
For
each dimension, assign a positive and negative.
Add the vectors in the same dimension.
|
 |
Resultant =
((3.54 cm)2 + (2.55 cm)2)
1/2
Resultant = 4.36 cm
Angle between North and Vector
2.55 cm
Angle = Sin -1 ( ------------- )
4.36 cm
Angle = 35.8o
Heading = 0.0o + 35.8o
Heading = 35.8o
Resultant = 4.36 cm at 35.8o
|
Draw
the two components and solve for the resultant vector using
hypotenuse = (a2 + b2)
1/2
Solve for one of the two angles and convert
into a heading.
opposite
Angle = Sin -1 ( ----------------- )
hypotenuse
|
|
Table of Contents
|
