Vector Refresher
Part 2
• Vector Addition
• Multiplication by Scalar
Quantities
• Graphical Representations
• Analytical Methods
Vector Addition
• A lot of times, it’s important to know the resultant of a
set of vectors
• The resultant vector can be described as the sum of all
the vector quantities you wish to evaluate
• We’ll look at graphical and analytical techniques for
doing this
Graphical Methods
For Vector Addition
• 2 ways to look at vector addition graphically
• Parallelogram method
A
B
Connect the tails of both vectors to
start.
Graphical Methods
For Vector Addition
• 2 ways to look at vector addition graphically
• Parallelogram method
A
B
Create a parallelogram with each
vector as one of the sides.
Graphical Methods
For Vector Addition
• 2 ways to look at vector addition graphically
• Parallelogram method
A
B
The resultant goes from the place
where 2 tails meet to the place where 2
heads meet
A+B
Graphical Methods
For Vector Addition
• 2 ways to look at vector addition graphically
• Head-to-tail method
A
B
With this method, the vectors are
added by placing the tail of a vector at
the head of another one to make a
continuous path.
Graphical Methods
For Vector Addition
• 2 ways to look at vector addition graphically
• Head-to-tail method
A
B
The addition of these vectors results in
a vector that starts at the tail of the first
vector and ends at the head of the final
vector
A+B
Graphical Methods
For Vector Addition
• 2 ways to look at vector addition graphically
• Head-to-tail method
B
A
C
If we had a 3rd vector, we could add it
to the previous result
A+B
Graphical Methods
For Vector Addition
• 2 ways to look at vector addition graphically
• Head-to-tail method
B
A
C
If we had a 3rd vector, we could add it
to the previous result
A+B
A+ B+C
Multiplying a Vector
By a Scalar
When a vector is multiplied by a scalar, the length of the
vector can be elongated or shortened. If the scalar is a
negative number, the direction is reversed.
Multiplying a Vector
By a Scalar
When a vector is multiplied by a scalar, the length of the
vector can be elongated or shortened. If the scalar is a
negative number, the direction is reversed.
A
2A
Multiplying a vector by
2 yields a vector with
the SAME
DIRECTION and
twice the length
Multiplying a Vector
By a Scalar
When a vector is multiplied by a scalar, the length of the
vector can be elongated or shortened. If the scalar is a
negative number, the direction is reversed.
A
1
A
2
Multiplying a vector by
1/2 yields a vector
with the SAME
DIRECTION and half
the length
Multiplying a Vector
By a Scalar
When a vector is multiplied by a scalar, the length of the
vector can be elongated or shortened. If the scalar is a
negative number, the direction is reversed.
A
-A
Multiplying a vector by
-1 yields a vector with
the OPPOSITE
DIRECTION and the
same length
Example
1
Draw A - 3B
2
Start by multiplying vector A by
1/2
A
B
Example
1
Draw A - 3B
2
Start by multiplying vector A by
1/2
1
A
2
A
B
Example
1
Draw A - 3B
2
Next, multiply vector B by -3
- 3B
1
A
2
A
B
Example
1
Draw A - 3B
2
Now add these vectors together
- 3B
1
A
2
A
B
Example
1
Draw A - 3B
2
A
The resultant vector is shown in
green
- 3B
1
A
2
1
A - 3B
2
B
Analytical Method
for Vector Addition
When vectors are added to each other, the components in
each direction are summed.
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
U +V = (a + d)iˆ + (b + e) jˆ + (c + f )kˆ
Analytical Method
for Vector Addition
When vectors are added to each other, the components in
each direction are summed.
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
U +V = (a + d)iˆ + (b + e) jˆ + (c + f )kˆ
Analytical Method
for Vector Addition
When vectors are added to each other, the components in
each direction are summed.
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
U +V = (a + d)iˆ + (b + e) jˆ + (c + f )kˆ
Multiplying a Vector
By a Scalar
When multiplying a vector by a scalar, the scalar must be
applied to each component of the vector
Thus, if we multiply V
we get:
= aiˆ + bjˆ + ckˆ by a scalar “k”,
kV = kaiˆ + kbjˆ + kckˆ
Example
Determine the resultant force caused by the following 3
force vectors.
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs
1
F3 = 2F1 - F2
3
The first thing to do is determine F3
(
2F1 = 2 éë16iˆ +10 jˆ + 2kˆùûlbs
)
Example
Determine the resultant force caused by the following 3
force vectors.
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs
1
F3 = 2F1 - F2
3
The first thing to do is determine F3
(
2F1 = 2 éë16iˆ +10 jˆ + 2kˆùûlbs
)
2F1 = éë(2·16)iˆ + (2·10) jˆ + (2·2)kˆùûlbs
Example
Determine the resultant force caused by the following 3
force vectors.
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs
1
F3 = 2F1 - F2
3
The first thing to do is determine F3
(
2F1 = 2 éë16iˆ +10 jˆ + 2kˆùûlbs
)
2F1 = éë(2·16)iˆ + (2·10) jˆ + (2·2)kˆùûlbs
2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs
Example
Determine the resultant force caused by the following 3
force vectors.
The first thing to do is determine F3
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs
1
F3 = 2F1 - F2
3
1
1 é ˆ
F2 = ë6i + 6 jˆ + 9kˆùûlbs
3
3
(
)
Example
Determine the resultant force caused by the following 3
force vectors.
The first thing to do is determine F3
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs
1
F3 = 2F1 - F2
3
1
1 é ˆ
F2 = ë6i + 6 jˆ + 9kˆùûlbs
3
3
éæ 1 ö ˆ æ 1 ö ˆ æ 1 ö ˆù
1
F2 = êç ·6 ÷ i + ç ·6 ÷ j + ç ·9 ÷ k úlbs
3
ëè 3 ø è 3 ø è 3 ø û
(
)
Example
Determine the resultant force caused by the following 3
force vectors.
The first thing to do is determine F3
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs
1
F3 = 2F1 - F2
3
1
1 é ˆ
F2 = ë6i + 6 jˆ + 9kˆùûlbs
3
3
éæ 1 ö ˆ æ 1 ö ˆ æ 1 ö ˆù
1
F2 = êç ·6 ÷ i + ç ·6 ÷ j + ç ·9 ÷ k úlbs
3
ëè 3 ø è 3 ø è 3 ø û
1
F2 = éë2iˆ + 2 jˆ + 3kˆùûlbs
3
(
)
Example
Determine the resultant force caused by the following 3
force vectors.
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs
1
F3 = 2F1 - F2
3
The first thing to do is determine F3
2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs
1
F2 = éë2iˆ + 2 jˆ + 3kˆùûlbs
3
1
2F1 - F2 = éë(32 - 2)iˆ + (20 - 2) jˆ + (4 - 3)kˆùûlbs
3
1
2F1 - F2 = éë30iˆ +18 jˆ + kˆùûlbs
3
F3 = éë30iˆ +18 jˆ + kˆùûlbs
Example
Determine the resultant force caused by the following 3
force vectors.
Now we can add the 3 vectors together
to determine the resultant force
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs
F3 = éë30iˆ +18 jˆ + kˆùûlbs
FR = F1 + F2 + F3 = (16 + 6 + 30)iˆ
Example
Determine the resultant force caused by the following 3
force vectors.
Now we can add the 3 vectors together
to determine the resultant force
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs
F3 = éë30iˆ +18 jˆ + kˆùûlbs
FR = F1 + F2 + F3 = (16 + 6 + 30)iˆ + (10 + 6 +18) jˆ
Example
Determine the resultant force caused by the following 3
force vectors.
Now we can add the 3 vectors together
to determine the resultant force
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs
F3 = éë30iˆ +18 jˆ + kˆùûlbs
FR = F1 + F2 + F3 = (16 + 6 + 30)iˆ + (10 + 6 +18) jˆ + (2 + 9 +1)kˆ
Example
Determine the resultant force caused by the following 3
force vectors.
Don’t forget the units, they’re as
important to the answer as the numbers
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
are
F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs
F3 = éë30iˆ +18 jˆ + kˆùûlbs
FR = F1 + F2 + F3 = éë(16 + 6 + 30)iˆ + (10 + 6 +18) jˆ + (2 + 9 +1)kˆùûlbs
Example
Determine the resultant force caused by the following 3
force vectors.
Don’t forget the units, they’re as
important to the answer as the numbers
F1 = éë16iˆ +10 jˆ + 2kˆùûlbs
are
F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs
F3 = éë30iˆ +18 jˆ + kˆùûlbs
FR = F1 + F2 + F3 = éë(16 + 6 + 30)iˆ + (10 + 6 +18) jˆ + (2 + 9 +1)kˆùûlbs
FR = éë52iˆ + 34 jˆ +12kˆùûlbs