geometric vectors
geometric vectors
Methods of Adding Vectors Geometrically
MCV4U: Calculus & Vectors
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.
Adding and Subtracting Vectors
This makes it possible for us to construct diagrams that
represent vector addition or subtraction of two or more
vectors.
J. Garvin
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geometric vectors
geometric vectors
Methods of Adding Vectors Geometrically
Methods of Adding Vectors Geometrically
Triangle Method of Vector Addition
Example
~ and BC
~ , arranged head to tail as
Given two vectors, AB
~ is the sum of AB
~ + BC
~ .
shown below, the resultant AC
Given vectors ~a and ~b, draw ~a + ~b.
Using the triangle method of vector addition,
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geometric vectors
geometric vectors
Methods of Adding Vectors Geometrically
Methods of Adding Vectors Geometrically
Parallelogram Method of Vector Addition
Example
~ and AD,
~ arranged tail-to-tail as
Given two vectors, AB
~ = AD
~ and DC
~ = AB.
~ The resultant AC
~ is
shown, let BC
~ + BC
~ or AD
~ + DC
~ .
the sum of AB
Given vectors ~a and ~b, draw ~a + ~b.
Using the parallelogram method of vector addition,
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geometric vectors
geometric vectors
Methods of Subtracting Vectors Geometrically
Methods of Subtracting Vectors Geometrically
Tail-to-Tail Method of Vector Subtraction
Example
~ and AC
~ , arranged tail-to-tail as
Given two vectors, AB
~ is the difference of AC
~ − AB.
~
shown, the resultant BC
Given vectors ~a and ~b, draw ~a − ~b.
Using the tail-to-tail method of vector subtraction,
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geometric vectors
geometric vectors
Methods of Subtracting Vectors Geometrically
Adding and Subtracting Vectors
Alternatively, a vector may be subtracted from another using
its opposite vector.
Example
Opposite Vector Method of Vector Subtraction
~ + BC
~ as a single
Using the following diagram, express AB
vector.
~ and AC
~ , arranged tail to tail as
Given two vectors, AB
~ = −AB
~ = BA.
~ The resultant AD
~ is the
shown, let CD
~ − AB.
~
difference of AC
~ + BC
~ = AC
~
AB
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geometric vectors
geometric vectors
Adding and Subtracting Vectors
Adding and Subtracting Vectors
Example
Example
~ − CB
~ as a single
Using the following diagram, express DB
vector.
~ + CD)
~ + DA
~ as a
Using the following diagram, express (BC
single vector.
~ − CB
~ = DB
~ + BC
~ = DC
~
DB
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~ + CD)
~ + DA
~ = BD
~ + DA
~ = BA
~
(BC
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geometric vectors
geometric vectors
Adding and Subtracting Vectors
Adding and Subtracting Vectors
Example
The last example illustrates the associative property of vector
addition.
~ + (CD
~ + DA)
~ as a
Using the following diagram, express BC
single vector.
Properties of Vector Addition and Subtraction
~:
Given vectors ~u , ~v and w
• (~u + ~
~ = ~u + (~v + w
~ ) (associative property)
v) + w
• ~u + ~
v = ~v + ~u (commutative property)
• ~
v + ~0 = ~v (identity property)
The zero vector, ~0, has a magnitude of zero and arbitrary
direction. Thus, adding a vector to the zero vector results in
the original vector.
~ + (CD
~ + DA)
~ = BC
~ + CA
~ = BA
~
BC
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geometric vectors
geometric vectors
Adding and Subtracting Vectors
Adding and Subtracting Vectors
Example
Example
~ = ~x and BC
~ = ~y .
Using the following diagram, let AB
~ in terms of ~x and ~y .
Express EF
~ = ~x and BC
~ = ~y .
Using the following diagram, let AB
~ in terms of ~x and ~y .
Express BG
~ = CB
~ = −BC
~ = −~y
EF
~ = BC
~ + CG
~ = BC
~ + BA
~ = BC
~ − AB
~ = ~y − ~x
BG
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geometric vectors
geometric vectors
Adding and Subtracting Vectors
Adding and Subtracting Vectors
Example
Example
~ = ~x and BC
~ = ~y .
Using the following diagram, let AB
~ in terms of ~x and ~y .
Express AD
A ship travels 150 km due east of port, then assumes a
bearing of N50◦ E for 100 km. Use trigonometry to determine
the displacement of the ship, and its direction.
Use the following diagram.
~ = AB
~ + BC
~ + CD
~ = AB
~ + BC
~ + BG
~ = ~x +~y +~y −~x = 2~y
AD
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geometric vectors
geometric vectors
Adding and Subtracting Vectors
Adding and Subtracting Vectors
The displacement is |~r |, where r is the resultant vector. Use
the cosine law.
q
|~r | = |~u |2 + |~v |2 − 2|~u ||~v | cos R
p
= 1502 + 1002 − 2 · 150 · 100 cos 140◦
The direction can be found if we know the measure of ∠V .
Use the sine law.
≈ 235.5km
sin V
sin R
=
|~v |
|~r |
∠V ≈ sin−1
≈ 16◦
100 · sin 140◦
235.5
The displacement is approximately 235.5 km, at a bearing of
approximately N74◦ E.
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geometric vectors
Questions?
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