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©2008 by W.H. Freeman and Company
Vector Notation and Operations
Mark LesmeisterDawson High School
Vectors: 5 Minute Review
• A vector is a quantity that has both a magnitude and a direction.– Examples include position, displacement, velocity,
acceleration and force.• A vector can be expressed by its components.– Components are the perpendicular vectors that
add up to the original vector.– Use sine and cosine to find components.
Vectors: 5 Minute Review
• Vectors can be multiplied by scalars.– Multiplying a vector by a scalar changes the
magnitude but leaves the direction unchanged.
V
2V
Vectors: 5 Minute Review
• Vectors can be added or subtracted.– To add vectors graphically, draw one after the
other, tip to tail.
©2008 by W.H. Freeman and Company
Vectors: 5 Minute Review
• Vectors can be added or subtracted.– To add vectors graphically, draw one after the
other, tip to tail.– To add vectors algebraically,• Resolve the vectors into components.• Add the components of each direction.• Use the Pythagorean Theorem to find the magnitude of
the resultant vector and inverse tan to find the angle.
Vector Addition Example
• A boy pushes horizontally on a wagon with a force of 10 N. A girl pulls on the wagon’s handle at an angle of 30 degrees from the horizontal with a force of 8 N. What is the net force acting on the wagon?
Unit Vectors
• A unit vector is a vector that points along the x, y or z axis and is one unit long.– The symbols for the x, y, and z unit vectors are
– Any vector can be expressed as a sum of its x, y and z components multiplied by unit vectors.
– The dimensions of the quantity are stated along with the unit vectors, e.g.
k j,i ˆand ,ˆˆ
jiv
m/s 2m/s 3
©2008 by W.H. Freeman and Company
Unit Vector Example
• A car has a velocity of 6 m/s in a direction 30o
north of east. Express this vector in terms of unit vectors. Let east be the positive x direction and north the positive y direction.
Operations with Vectors: Magnitude
• To find the magnitude of a vector, use the Pythagorean Theorem.
222then
If
zyxr
zyx
r
kjir
Operations with vectors: Multiplying by a scalar
• To multiply a vector by a scalar, multiply each component by the scalar.
jiF
jiF
jia
aF
)N (6 )N (9
)m/s 2 kg (3 )m/s 3 kg (3
then kg, 3 and )m/s (2 )m/s 3( If22
22
m
m
kjir
kjir
CzCyCxC
zyxC
then
andconstant a is If
Operations with Vectors: Addition and subtraction
• To add two vectors, add their components.
kjivvv
kjvjiv
21r
21
563
54 and 23
Operations with Vectors: Dot Product
• The dot product is the product of the magnitude of two vectors times the cos of the angle between them.
• The cos means that only the component of one vector that lies along the direction of the other vector contributes to the product.
• Because the dot product is a scalar, it is sometimes called the scalar product.
cosABBA
Operations with Vectors: The Dot Product with Components
• The dot product in component notation is the sum of the products of the various components.
zzyyxx
zyxzyx
BABABA
BBBAAA
BA
kjiBkjiA
then
and Let
Operations with Vectors: The Cross Product
• The cross product yields a vector at right angles to the two vectors.
. ofdirection in the points now Your thumb
. ofdirection in the fingers
otheryour and ofdirection in the forefingeryour Point
:Rule HandRight by thegiven is ofdirection The
sin
BA
B
A
BA
BA
AB
Operations with Vectors: The Cross Product with Components
• Using the right hand rule, we can derive the following results for cross products of unit vectors:
kij
ijk
jki
jik
ikj
kji
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆ
i
j
k
Operations with Vectors: The Cross Product with Components
• To take the cross product using components, use the distributive property:
kjF
jir
ˆ2ˆ2
ˆ2ˆ3
)ˆ22()ˆ2ˆ2()ˆ2ˆ3()ˆ2ˆ3(
)ˆ2ˆ2()ˆ2ˆ3(
kjjjkiji
kjjiFr
ij6kFr 40ˆˆ6
Operations with Vectors: Derivatives
• A vector in component form is just a sum of vectors in the x, y and z directions. So, derivatives just follow the rule for sums: Find the derivative of each term separately.
• Ex.:jir
235 tt
jir
v tdt
d65
jrv
a
62
2
dt
d
dt
d
