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Vector basics handout - Lakehead University- Edit Course Discrete Mathematicsyyuyu

Lakehead University- Edit Course Discrete Mathematicsyyuyu
Course

Discrete Mathematics  (Mathematics 1271)

23 Documents
Students shared 23 documents in this course
Academic year: 2015/2016
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Lakehead University

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g e o m e t r i c v e c t o r s

MCV4U: Calculus & Vectors

Vector Basics

J. Garvin

Slide 1/

g e o m e t r i c v e c t o r s

Vectors vs. Scalars

Avectoris a quantity with bothmagnitudeanddirection.

Ascalaris a quantity that has onlymagnitude.

For example, speed is a scalar (80 km/h), while velocity is a

vector (75 km/h NE).

J. Garvin — Vector Basics

Slide 2/

g e o m e t r i c v e c t o r s

Vectors vs. Scalars

Example

Classify each quantity as a vector or a scalar.

1 The temperature outside is 3

◦ C (scalar)

2 Gravity causes a ball to accelerate at 9 m/s

2

downward (vector)

3 A student has a mass of 68 kg (scalar)

4 A car drives 25 km west (vector)

J. Garvin — Vector Basics

Slide 3/

g e o m e t r i c v e c t o r s

Vector Notation

Vectors are often represented using arrows, since arrows have

both a length (magnitude) and a direction.

The three vectors are

~

AB,

~

BCand

~

AC.

The magnitude of a vector is usually indicated using vertical

bars (absolute value).

|

~

AB|= 4 |

~

BC|= 3 |

~

AC|= 5

J. Garvin — Vector Basics

Slide 4/

g e o m e t r i c v e c t o r s

Representing Vectors

Vectors can be represented in many ways.

1 Using a diagram

2 Using words (250 km northeast)

3 Using symbols (

~

AB)

J. Garvin — Vector Basics

Slide 5/

g e o m e t r i c v e c t o r s

Terminology

Equivalent vectorsare those with the same magnitude and

direction (e. 9 m/s

2 down and 9 m/s

2 down).

Opposite vectorsare those with the same magnitude, but

have the opposite direction (e. 5 km NE and 5 km SW).

The vector−vis opposite tov.

Parallel vectorsmay have different magnitudes, but their

directions are either the same or opposite (e. 3 N left, 15 N

right).

Theangle between two vectors,θ, is the acute or obtuse

angle formed between them when drawn tail-to-tail.

J. Garvin — Vector Basics

Slide 6/

g e o m e t r i c v e c t o r s

Terminology

Example

For

~

ABbelow, state two equivalent vectors, two opposite

vectors, and two parallel vectors.

Equivalent:

~

FG,

~

GCand

~

ED.

Opposite:

~

CG,

~

GF,

~

DEand

~

BA.

Parallel: all seven above, plus

~

FCand

~

CF.

J. Garvin — Vector Basics

Slide 7/

g e o m e t r i c v e c t o r s

Bearings

True bearingsrotate clockwise, beginning at the north.

Quadrant bearingsmeasure the angle east or west of the

north-south line.

J. Garvin — Vector Basics

Slide 8/

g e o m e t r i c v e c t o r s

Bearings

Express the following vector using both a true bearing and

quadrant bearing.

The true bearing is 140

◦ , and the quadrant bearing is S

◦ E.

J. Garvin — Vector Basics

Slide 9/

g e o m e t r i c v e c t o r s

Unit Vectors

Aunit vector, ˆv, is a vector with a magnitude of 1. That is,

|vˆ|= 1.

Sincek·

1

k

= 1, a unit vector in the direction of any vector~v

can be found by multiplying~vby the reciprocal of its

magnitude.

vˆ=

1

|~v|

·~vor ˆv=

~v

|~v|

Any vector,~v, can be expressed as the product of its

magnitude,|~v|and a unit vector, ˆv.

v= ˆv· |v|

J. Garvin — Vector Basics

Slide 10/

g e o m e t r i c v e c t o r s

Unit Vectors

Example

Vector~vhas a magnitude of

5

2

units, with a bearing of 35

◦ .

Determine two unit vectors parallel to

~

v.

Since

5

2

·

2

5

= 1, vector~a=

2

5

vis a unit vector parallel tov.

Since−vis parallel tov,

~

b=−

2

5

~vis another unit vector

parallel to~v.

J. Garvin — Vector Basics

Slide 11/

g e o m e t r i c v e c t o r s

Questions?

J. Garvin — Vector Basics

Slide 12/

Was this document helpful?

Vector basics handout - Lakehead University- Edit Course Discrete Mathematicsyyuyu

Course: Discrete Mathematics  (Mathematics 1271)

23 Documents
Students shared 23 documents in this course

University: Lakehead University

Was this document helpful?
g e o m e t r i c v e c t o r s
MCV4U: Calculus & Vectors
Vector Basics
J. Garvin
Slide 1/12
g e o m e t r i c v e c t o r s
Vectors vs. Scalars
Avector is a quantity with both magnitude and direction.
Ascalar is a quantity that has only magnitude.
For example, speed is a scalar (80 km/h), while velocity is a
vector (75 km/h NE).
J. Garvin — Vector Basics
Slide 2/12
g e o m e t r i c v e c t o r s
Vectors vs. Scalars
Example
Classify each quantity as a vector or a scalar.
1The temperature outside is 3◦C (scalar)
2Gravity causes a ball to accelerate at 9.8 m/s2
downward (vector)
3A student has a mass of 68 kg (scalar)
4A car drives 25 km west (vector)
J. Garvin — Vector Basics
Slide 3/12
g e o m e t r i c v e c t o r s
Vector Notation
Vectors are often represented using arrows, since arrows have
both a length (magnitude) and a direction.
The three vectors are ~
AB,~
BC and ~
AC .
The magnitude of a vector is usually indicated using vertical
bars (absolute value).
|~
AB|= 4 |~
BC |= 3 |~
AC |= 5
J. Garvin — Vector Basics
Slide 4/12
g e o m e t r i c v e c t o r s
Representing Vectors
Vectors can be represented in many ways.
1Using a diagram
2Using words (250 km northeast)
3Using symbols ( ~
AB)
J. Garvin — Vector Basics
Slide 5/12
g e o m e t r i c v e c t o r s
Terminology
Equivalent vectors are those with the same magnitude and
direction (e.g. 9.8 m/s2down and 9.8 m/s2down).
Opposite vectors are those with the same magnitude, but
have the opposite direction (e.g. 5 km NE and 5 km SW).
The vector −~
vis opposite to ~
v.
Parallel vectors may have different magnitudes, but their
directions are either the same or opposite (e.g. 3 N left, 15 N
right).
The angle between two vectors,θ, is the acute or obtuse
angle formed between them when drawn tail-to-tail.
J. Garvin — Vector Basics
Slide 6/12

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