Download Geometric Vector Addition and Subtraction and more Study notes Calculus in PDF only on Docsity!
MCV4U: Calculus & Vectors
Adding and Subtracting Vectors
J. Garvin
Slide 1/
Methods of Adding Vectors Geometrically
Recall that two vectors are equivalent if they have the same
magnitude and direction.
This means that vectors can change their positions and
remain equivalent, as long as they maintain their magnitudes
and directions.
This makes it possible for us to construct diagrams that
represent vector addition or subtraction of two or more
vectors.
J. Garvin โ Adding and Subtracting Vectors
Slide 2/
g e o m e t r i c v e c t o r s
Methods of Adding Vectors Geometrically
Triangle Method of Vector Addition
Given two vectors,
AB and
BC , arranged head to tail as
shown below, the resultant
AC is the sum of
AB +
BC.
J. Garvin โ Adding and Subtracting Vectors
Slide 3/
g e o m e t r i c v e c t o r s
Methods of Adding Vectors Geometrically
Example
Given vectors ~a and
b, draw ~a +
b.
Using the triangle method of vector addition,
J. Garvin โ Adding and Subtracting Vectors
Slide 4/
g e o m e t r i c v e c t o r s
Methods of Adding Vectors Geometrically
Parallelogram Method of Vector Addition
Given two vectors,
AB and
AD, arranged tail-to-tail as
shown, let
BC =
AD and
DC =
AB. The resultant
AC is
the sum of
AB +
BC or
AD +
DC.
J. Garvin โ Adding and Subtracting Vectors
g e o m e t r i c v e c t o r s
Methods of Adding Vectors Geometrically
Example
Given vectors
a and
b, draw
a +
b.
Using the parallelogram method of vector addition,
J. Garvin โ Adding and Subtracting Vectors
Methods of Subtracting Vectors Geometrically
Tail-to-Tail Method of Vector Subtraction
Given two vectors,
AB and
AC , arranged tail-to-tail as
shown, the resultant
BC is the difference of
AC โ
AB.
J. Garvin โ Adding and Subtracting Vectors
Slide 7/
Methods of Subtracting Vectors Geometrically
Example
Given vectors ~a and
b, draw ~a โ
b.
Using the tail-to-tail method of vector subtraction,
J. Garvin โ Adding and Subtracting Vectors
Slide 8/
g e o m e t r i c v e c t o r s
Methods of Subtracting Vectors Geometrically
Alternatively, a vector may be subtracted from another using
its opposite vector.
Opposite Vector Method of Vector Subtraction
Given two vectors,
AB and
AC , arranged tail to tail as
shown, let
CD = โ
AB =
BA. The resultant
AD is the
difference of
AC โ
AB.
J. Garvin โ Adding and Subtracting Vectors
Slide 9/
g e o m e t r i c v e c t o r s
Adding and Subtracting Vectors
Example
Using the following diagram, express
AB +
BC as a single
vector.
AB +
BC =
AC
J. Garvin โ Adding and Subtracting Vectors
Slide 10/
g e o m e t r i c v e c t o r s
Adding and Subtracting Vectors
Example
Using the following diagram, express
DB โ
CB as a single
vector.
DB โ
CB =
DB +
BC =
DC
J. Garvin โ Adding and Subtracting Vectors
g e o m e t r i c v e c t o r s
Adding and Subtracting Vectors
Example
Using the following diagram, express (
BC +
CD) +
DA as a
single vector.
BC +
CD) +
DA =
BD +
DA =
BA
J. Garvin โ Adding and Subtracting Vectors
Adding and Subtracting Vectors
The displacement is |~r |, where r is the resultant vector. Use
the cosine law.
r | =
u|
2
v |
2 โ 2 |
u||
v | cos R
2
2 โ 2 ยท 150 ยท 100 cos 140
โฆ
โ 235. 5 km
J. Garvin โ Adding and Subtracting Vectors
Slide 19/
Adding and Subtracting Vectors
The direction can be found if we know the measure of โ V.
Use the sine law.
sin V
|~v |
sin R
|~r |
โ V โ sin
โ 1
100 ยท sin 140
โฆ
โฆ
The displacement is approximately 235.5 km, at a bearing of
approximately N
โฆ E.
J. Garvin โ Adding and Subtracting Vectors
Slide 20/
g e o m e t r i c v e c t o r s
Questions?
J. Garvin โ Adding and Subtracting Vectors
Slide 21/
