Vector Refresher
Part 1
• Definition
• Component Notation
• Making a Vector
• Calculating the Magnitude
• Calculating the Direction
What is a Vector?
• A vector is a quantity that has the following characteristics
• Magnitude (size)
• Direction
• May have units
What is a Vector?
• A vector is a quantity that has the following characteristics
• Magnitude (size)
• Direction
• May have units
• For now, we’ll use an arrow on top to denote vectors, for
example the vector “V” will be expressed as V
What is a Vector?
• A vector is a quantity that has the following characteristics
• Magnitude (size)
• Direction
• May have units
• For now, we’ll use an arrow on top to denote vectors, for
example the vector “V” will be expressed as V
• A vector appears as a set of components relative to a
coordinate system. We’ll largely use a Cartesian
Coordinate System for reference. Thus, in 3 dimensions,
the vector V will have components in the x, y, and z
directions
Vector V.s. Scalar
• A vector is something that has a size and a direction
Vector V.s. Scalar
• A vector is something that has a size and a direction
• A scalar is something with size
Vector V.s. Scalar
• A vector is something that has a size and a direction
• A scalar is something with size
• Speed is a scalar quantity (55 mph)
Vector V.s. Scalar
• A vector is something that has a size and a direction
• A scalar is something with size
• Speed is a scalar quantity (55 mph)
• Velocity is a vector quantity (55 mph due East)
Anatomy of a
Vector
a vector has 2 main pieces
Anatomy of a
Vector
a vector has 2 main pieces
A
(a,b,c)
The tail is where the vector
starts from
Anatomy of a
Vector
a vector has 2 main pieces
A
B
(a,b,c)
(d,e,f)
The tail is where the vector
starts from
The head is where it ends up.
Anatomy of a
Vector
a vector has 2 main pieces
A
(a,b,c)
The tail is where the vector
starts from
RAB
B
(d,e,f)
The head is where it ends up.
A given vector that goes FROM point A TO point B will be
denoted with a subscript ‘AB’. For example, if the vector above
could be called RAB
Component
Notation
• A typical vector, V, will appear in the following form:
V = aiˆ + bjˆ + ckˆ
Component
Notation
• A typical vector, V, will appear in the following form:
V = aiˆ + bjˆ + ckˆ
• The
iˆ term denotes the component in the x direction ( a )
Component
Notation
• A typical vector, V, will appear in the following form:
V = aiˆ + bjˆ + ckˆ
• The
• The
iˆ term denotes the component in the x direction ( a )
jˆ term denotes the component in the y direction ( b )
Component
Notation
• A typical vector, V, will appear in the following form:
V = aiˆ + bjˆ + ckˆ
• The
• The
• The
iˆ term denotes the component in the x direction ( a )
jˆ term denotes the component in the y direction ( b )
kˆ term denotes the component in the z direction ( c )
A Typical Vector
• A typical vector in 2D will look like this
y
(a,b)
b
V = aiˆ + bjˆ
θ
a
x
A Typical Vector
• A typical vector in 3D will look like this
z
c
V = aiˆ + bjˆ + ckˆ
(a,b,c)
b
a
x
y
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can be found
using trigonometry if θ
is given
(a,b)
b
V = aiˆ + bjˆ
θ
a
x
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can be found
using trigonometry if θ
is given
(a,b)
b
V = aiˆ + bjˆ
θ
a
a = lengthcos(q )
x
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can be found
using trigonometry if θ
is given
(a,b)
b
V = aiˆ + bjˆ
θ
a
a = lengthcos(q )
b = lengthsin(q )
x
A Typical Vector
A typical vector in 2D will look like this:
y
Conversely, if
components a and b are
know,θcan be found
(a,b)
b
V = aiˆ + bjˆ
θ
a
æbö
q = tan ç ÷
èaø
-1
x
A Typical Vector
A typical vector in 2D will look like this:
y
a = lengthcos(q )
(a,b)
b
b = lengthsin(q )
V = aiˆ + bjˆ
æbö
q = tan ç ÷
èaø
-1
θ
a
x
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can also be found
if the slope is given
(a,b)
b
V = aiˆ + bjˆ
d
c
a
x
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can also be found
if the slope is given
(a,b)
b
V = aiˆ + bjˆ
hypotenuse = c + d
2
d
c
a
x
2
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can also be found
if the slope is given
(a,b)
b
V = aiˆ + bjˆ
hypotenuse = c + d
2
æ
ö
c
a = length ç
÷
2
2
è c +d ø
d
c
a
2
x
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can also be found
if the slope is given
(a,b)
b
V = aiˆ + bjˆ
hypotenuse = c + d
2
d
c
a
x
2
æ
ö
c
a = length ç
÷
2
2
è c +d ø
æ
ö
d
b = length ç
÷
è c2 + d 2 ø
A Typical Vector
A typical vector in 2D will look like this:
y
If the slope is given, we
can find θ
(a,b)
b
ædö
q = tan ç ÷
ècø
V = aiˆ + bjˆ
-1
d
c
θ
a
x
A Typical Vector
A typical vector in 2D will look like this:
y
(a,b)
b
V = aiˆ + bjˆ
d
c
θ
a
x
æ
ö
c
a = length ç
÷
2
2
è c +d ø
æ
ö
d
b = length ç
÷
è c2 + d 2 ø
-1 æ d ö
q = tan ç ÷
ècø
Vector
Construction
A vector can be constructed if you know the initial point and
end point of the vector
(d,e,f)
(a,b,c)
Vector
Construction
A vector can be constructed if you know the initial point and
end point of the vector
The vector is found by subtracting the starting point from the
end point
(d,e,f)
(a,b,c)
Vector
Construction
A vector can be constructed if you know the initial point and
end point of the vector
The vector is found by subtracting the starting point from the
end point
V = (d - a)iˆ + (e - b) jˆ + ( f - c)kˆ
(a,b,c)
(d,e,f)
Magnitude of a
Vector
• Often times, we need to know the magnitude of a
given vector (how long the arrow is)
• This will be denoted as:
V
Calculating The
Magnitude of a Vector
The magnitude is the square root of the sum of the
squared components
V = aiˆ + bjˆ + ckˆ
V = a +b +c
2
2
2
Calculating The
Magnitude of a Vector
The magnitude is the square root of the sum of the
squared components
V = aiˆ + bjˆ + ckˆ
V = a +b +c
2
2
2
This calculation yields a SCALAR value, thus the
magnitude of a vector is a SCALAR quantity, that has
no associated direction.
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can be found
using trigonometry if θ
is given
(a,b)
b
V = aiˆ + bjˆ
a = V cos(q )
b = V sin(q )
θ
a
x
A Typical Vector
A typical vector in 2D will look like this:
y
Components of a 2D
vector can also be found
if the slope is given
(a,b)
b
V = aiˆ + bjˆ
d
c
a
x
æ
ö
c
a= V ç
÷
è c2 + d 2 ø
æ
ö
d
b= V ç
÷
è c2 + d 2 ø
Unit Vectors
• Similar to how the magnitude describes only the size
of a vector, the unit vector describes only the direction
of a vector
• The unit vector is denoted as follows: uˆ
Unit Vectors
• Similar to how the magnitude describes only the size
of a vector, the unit vector describes only the direction
of a vector
• The unit vector is denoted as follows: uˆ
• Sometimes, this notation is accompanied by a subscript
that denotes the vector whose direction a unit vector
describes.
• The unit vector describing vector “V” could be
expressed as: uˆV
Calculating the
Unit Vector
• The unit vector is described as a vector divided by its
magnitude
V
uˆV =
V
• The magnitude of a unit vector is always 1 (hence the
name)
• This is a good way to check your work
Unit Vectors
If we have V = aiˆ + bjˆ,
then, the vector’s magnitude is V =
a +b
2
2
Unit Vectors
If we have V = aiˆ + bjˆ,
then, the vector’s magnitude is V =
and the unit vector is uˆV =
a
a2 + b2
iˆ +
a +b ,
2
2
b
a2 + b2
jˆ
Unit Vectors
If we have V = aiˆ + bjˆ,
then, the vector’s magnitude is V =
and the unit vector is uˆV =
a
a2 + b2
iˆ +
a +b ,
2
2
b
a2 + b2
NOTE: The unit vector will always be unitless.
jˆ
Vector Definition
• Another definition for a vector is its magnitude
multiplied by the direction it’s going.
V = V uˆV
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
(2,3,7)
(0,0,1)
x
y
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
The first step is to create a a vector, then
we’ll calculate the magnitude of it and
use that result to find the unit vector
(2,3,7)
(0,0,1)
x
y
V = éë(2 - 0)iˆ + (3- 0) jˆ + (7 -1)kˆùû ft
V = éë(2)iˆ + (3) jˆ + (6)kˆùû ft
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
We can find the magnitude of the vector
by squaring each component, adding
them, and taking the square root of that
addition
(2,3,7)
(0,0,1)
x
y
V = (2 ft)2 + (3 ft)2 + (6 ft)2
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
We can find the magnitude of the vector
by squaring each component, adding
them, and taking the square root of that
addition
(2,3,7)
(0,0,1)
y
V = (2 ft)2 + (3 ft)2 + (6 ft)2
V = 4 ft 2 + 9 ft 2 + 36 ft 2
x
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
We can find the magnitude of the vector
by squaring each component, adding
them, and taking the square root of that
addition
(2,3,7)
(0,0,1)
y
V = (2 ft)2 + (3 ft)2 + (6 ft)2
V = 4 ft 2 + 9 ft 2 + 36 ft 2
x
V = 49 ft 2
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
We can find the magnitude of the vector
by squaring each component, adding
them, and taking the square root of that
addition
(2,3,7)
(0,0,1)
y
V = (2 ft)2 + (3 ft)2 + (6 ft)2
V = 4 ft 2 + 9 ft 2 + 36 ft 2
x
V = 49 ft 2
V = 7 ft
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
The unit vector is found by dividing the
vector by its magnitude
(2,3,7)
(0,0,1)
x
y
2 ft ˆ 3 ft ˆ 6 ft ˆ
uˆV =
i+
j+
k
7 ft
7 ft
7 ft
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
Notice that the units cancel out
(2,3,7)
(0,0,1)
x
y
2 ft ˆ 3 ft ˆ 6 ft ˆ
uˆV =
i+
j+
k
7 ft
7 ft
7 ft
2ˆ 3ˆ 6 ˆ
uˆV = i + j + k
7 7
7
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
As a quick check, let’s confirm that the
magnitude of the unit vector is 1
2
(2,3,7)
(0,0,1)
x
æ2ö æ 3ö æ6ö
uˆV = ç ÷ + ç ÷ + ç ÷
è 7ø è 7ø è 7ø
y
2
2
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
As a quick check, let’s confirm that the
magnitude of the unit vector is 1
2
(2,3,7)
(0,0,1)
x
æ2ö æ 3ö æ6ö
uˆV = ç ÷ + ç ÷ + ç ÷
è 7ø è 7ø è 7ø
4 9 36
uˆV =
+ +
49 49 49
y
2
2
Example Problem
A vector starts at point (0 ft,0 ft,1ft) and terminates at
point (2 ft,3 ft,7 ft). Determine the size of the vector and
the unit vector that describes it’s direction.
z
As a quick check, let’s confirm that the
magnitude of the unit vector is 1
2
(2,3,7)
(0,0,1)
x
æ2ö æ 3ö æ6ö
uˆV = ç ÷ + ç ÷ + ç ÷
è 7ø è 7ø è 7ø
4 9 36
uˆV =
+ +
49 49 49
49
uˆV =
=1
49
y
2
2