Slide 1

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 2

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 3

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 4

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 5

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 6

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 7

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 8

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 9

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 10

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 11

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

Home

Tip-to-Tail

Home

Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
Home

Cross Product

Home

Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

Home

Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

Home

Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

Home

Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

Home

References
 http://mathworld.wolfram.com/Vector.html

Home


Slide 12

Vectors
By Scott Forbes

QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.

Vectors









Definition of a Vector
Addition
Multiplication
Unit Vector
Zero Vector
Theorems
Laws
References

Definition of a Vector
 Point of Application
- A vector is a visual representation of a force.
The force needs to have a point on which it is
being applied.
 Direction
- A vector must have a specific direction defined
by either coordinates, or degrees.
 Magnitude
- The vector must have length, or magnitude.
Magnitude defines the amount of force applied
to the point of application.
Home

Addition
 Tip-to-tail
- Adding by connecting the tip of vector
A to the tail of vector B. Then draw a
new vector from the tail of vector A to
the tip of vector B, and that is the sum of
the two vectors.
Click here for a visual representation.

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Tip-to-Tail

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Multiplication
 Dot Product
- Multiply the corresponding components
of vectors A and B. For example vector A
= [ 3,2 ], vector B = [ 4,7 ].
A • B = (3 x 4) + (2 x 7) = 26
 Cross Product
- Set up vectors in cross product form.
 Scalar
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Cross Product

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Unit Vector
 A unit vector is denoted as a “hatted”
letter. An example, â.
 Converting using the norm
- To convert a vector into a unit vector,
divide by the normal vector.
= normal vector
= unit vector

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Zero Vector
 Has no magnitude
 Has no direction
 All components are equal to zero

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Theorems
 Two vectors are equivalent if they have
the same direction and magnitude.
 If a vector, denoted by A, has the same
magnitude but opposite direction as
vector B, then vector B can be shown as
-A.

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Laws
 Addition
- Commutative law: A+B = B+A
- Associative law: A+(B+C) = (A+B)+C
 Multiplication
- mA = Am
- (m+n)A = mA + nA
m and n are different scalar quantities

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References
 http://mathworld.wolfram.com/Vector.html

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