vector refresher part 1
DESCRIPTION
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?. - PowerPoint PPT PresentationTRANSCRIPT
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?
• For now, we’ll use an arrow on top to denote vectors, for example the vector “V” will be expressed as
• A vector is a quantity that has the following characteristics• Magnitude (size)• Direction• May have units
What is a Vector?
• For now, we’ll use an arrow on top to denote vectors, for example the vector “V” will be expressed as• 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 will have components in the x, y, and z directions
• A vector is a quantity that has the following characteristics• Magnitude (size)• Direction• May have units
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
The tail is where the vector starts from
(a,b,c)
A
Anatomy of a Vector a vector has 2 main pieces
The tail is where the vector starts from
The head is where it ends up.
(a,b,c) (d,e,f)
A B
Anatomy of a Vector a vector has 2 main pieces
The tail is where the vector starts from
The head is where it ends up.
(a,b,c) (d,e,f)
A B
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
Component Notation• A typical vector, , will appear in the
following form:
Component Notation• A typical vector, , will appear in the
following form:
• The term denotes the component in the x direction ( a )
Component Notation• A typical vector, , will appear in the
following form:
• The term denotes the component in the x direction ( a )
• The term denotes the component in the y direction ( b )
Component Notation• A typical vector, , will appear in the
following form:
• The term denotes the component in the x direction ( a )
• The term denotes the component in the y direction ( b )
• The term denotes the component in the z direction ( c )
A Typical Vector• A typical vector in 2D will look like this
b
ax
y
θ
(a,b)
A Typical Vector• A typical vector in 3D will look like this
(a,b,c)
x
y
z
ab
c
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can be found using trigonometry if θ is given
θ
(a,b)
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can be found using trigonometry if θ is given
θ
(a,b)
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can be found using trigonometry if θ is given
θ
(a,b)
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Conversely, if components a and b are know,θcan be found
θ
(a,b)
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
θ
(a,b)
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can also be found if the slope is given(a,b)
dc
c
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can also be found if the slope is given(a,b)
d
c
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can also be found if the slope is given(a,b)
d
c
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can also be found if the slope is given(a,b)
d
c
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
If the slope is given, we can find θ
(a,b)
d
θ
c
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
(a,b)
d
θ
Vector ConstructionA vector can be constructed if you know the initial point and end point of the vector
(a,b,c)
(d,e,f)
Vector ConstructionA 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
(a,b,c)
(d,e,f)
Vector ConstructionA 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
(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:
Calculating The Magnitude of a Vector
The magnitude is the square root of the sum of the squared components
Calculating The Magnitude of a Vector
The magnitude is the square root of the sum of the squared components
This calculation yields a SCALAR value, thus the magnitude of a vector is a SCALAR quantity, that has no associated direction.
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can be found using trigonometry if θ is given
θ
(a,b)
c
A Typical VectorA typical vector in 2D will look like this:
b
ax
y
Components of a 2D vector can also be found if the slope is given(a,b)
d
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:
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:• 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:
Calculating the Unit Vector
• The unit vector is described as a vector divided by its magnitude
• The magnitude of a unit vector is always 1 (hence the name)
• This is a good way to check your work
Unit VectorsIf we have ,then, the vector’s magnitude is
Unit VectorsIf we have ,then, the vector’s magnitude is ,and the unit vector is
Unit VectorsIf we have ,then, the vector’s magnitude is ,and the unit vector is NOTE: The unit vector will always be unitless.
Vector Definition• Another definition for a vector is its
magnitude multiplied by the direction it’s going.
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
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
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
We can find the magnitude of the vector by squaring each component, adding them, and taking the square root of that addition
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
We can find the magnitude of the vector by squaring each component, adding them, and taking the square root of that addition
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
We can find the magnitude of the vector by squaring each component, adding them, and taking the square root of that addition
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
We can find the magnitude of the vector by squaring each component, adding them, and taking the square root of that addition
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
The unit vector is found by dividing the vector by its magnitude
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
Notice that the units cancel out
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
As a quick check, let’s confirm that the magnitude of the unit vector is 1
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
As a quick check, let’s confirm that the magnitude of the unit vector is 1
Example ProblemA 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.
x
y
z
(0,0,1)
(2,3,7)
As a quick check, let’s confirm that the magnitude of the unit vector is 1