Download - Dot Product & Cross Product of two vectors
Dot Product & Cross Product of two vectors
Work done by a force
F
s
θ
θW = F s cosθ
= F · s
F
s
Dot product (Scalar product)
a · b = |a| |b| cosθ
= axbx + ayby + azbz
0o <θ<180o is the angle between vectors a and b a · c = |a| |c| cos90o = 0
a and c are perpendicular or orthogonal. a · d = |a| |d| cos 00 = |a| |d| a · a = |a| |a| cos 00 = |a|2
θ
b
a
c
d
Properties of Dot Product
Commutative property
a ·b = b·a
Distributive property
a · ( b + c ) = a ·b + b·c
Example
a = (1, 2, 4), b =(-1, 2, -1)
a · b = 1x(-1) + 2x2 + 4x(-1) = -1
Example
a = (0, 1, -1), b = (2, -1, 1)
a · b = 0x2 + 1x(-1) +(-1)x1 = -2
Example
j
k
i· =
· =j
· =
· =
· =
· =
k
i j
j k
i
i k
(1,0,0) ·(1,0,0) =1
(0,1,0) ·(0,1,0) =1
(0,0,1) ·(0,0,1) =1
(1,0,0) ·(0,1,0) =0
(0,1,0) ·(0,0,1) =0
(1,0,0) ·(0,0,1) =0
ki
j
x
y
z
1
1
1
Example
Find the angle between vectors a = (1, 1, -1) and b = (2, -1, 0)
a · b = 1x2 + 1x(-1) +(-1)x0 = 1
cos θ = =
=
ba
ba 222222 0)1(2)1(11
1
15
1
Example
A(2,1, 0), B(1, -1,1), C(0, 2, 1) are three points. Find the angles in the triangle ABC
α
β
θA
B
C
Example
a = α + +2 , b= +β - , c= - +γ Find the numbers α, β, γ which make the
vectors a, b and c mutually perpendicular.
i j k i j k i j k
Example
a = + +2 , b= + - Construct any vector perpendicular to a and b
i j k i j k
Direction Cosinesia
iaˆ
ˆcos
x
ja
jaˆ
ˆcos
y
ka
kaˆ
ˆcos
z
y
aaxzyx aaaa
)0,0,1(),,(
aayzyx aaaa
)0,1,0(),,(
aazzyx aaaa
)1,0,0(),,(
aa
z
x
θz
θx
θy
i j
k
aaaa zyx aaa
,,ˆ
zyx cos,cos,cos
Example
Find the direction cosines of the vector
k2-j2is ˆˆˆ
Example
Find the unit vector in the direction of the vector a=(3, 4, 1).
Direction Ratios of a straight line To determine the inclination of a straight line. Components of any vector s that is parallel to line.
s = p q rkji ˆˆˆ rqp
, ,
Direction Ratios of a straight line L: Line L
Example
(Two dimension) Find a set of direction ratios for the straight line y=2x+1.
Example
Find the equation for a straight line which passes though point(1, 0, -1) and has a set of direction ratios of (1, 2, 2).
Components of a vector a=(ax, ay, az)
ia ˆ
ja ˆ
ka ˆ ki
j
x
y
z
1
1
1
a
(ax, ay, az)·(1, 0,0)=ax
(ax, ay, az)·(0, 1,0)=ay
(ax, ay, az)·(0, 0,1)=az
Rotation of Axes in Two dimensions…
IJ
K
θ
x
y
X
Y
P(x, y), P(X, Y)
i
j
k
I
J = (cos(π/2+θ), sin(π/2+ θ)
= (-sin θ, cos θ)
= (cosθ, sinθ)
X = (x, y)·(cosθ, sin θ)
= xcos θ + ysin θ
Y = (x, y)·(-sinθ, cos θ)
= -xsin θ + ycos θ
Rotation of Axes in Three Dimension…
k
i j
x
y
z
a
I
J
K
Z
X
Y
a=(x, y, z) = x i+y j+zk in Oxyz
a = (?, ?, ?) in OXYZ
O
Rotation of Axes in Three Dimension…
k
i j
x
y
z
I
JK
Z
X
Y
In OXYZ,
I=(1, 0, 0)
In Oxyz,
I = (l1, m1, n1)
O
J=(0, 1, 0)K=(0, 0, 1)
J = (l2, m2, n2) K = (l3, m3, n3)
l1
m1
n1
Rotation of Axes in Three Dimension…
k
i j
x
y
z
I
JK
Z
X
Y
In xyz,
i=(1, 0, 0)
In OXYZ,
i= (l1, l2, l3)
O
j=(0, 1, 0)k=(0, 0, 1)
j = (m1, m2, m3) k = (n1, n2, n3)
l1
l2
l3
Rotation of axes
Oxyz OXYZ
i (1, 0, 0) (l1 , l2 , l3)
j (0, 1, 0) (m1 , m2 , m3)
k (0, 0, 1) (n1 , n2 , n3)
I (l1 , m1 , n1) (1, 0, 0)
J (l2 , m2 , n2) (0, 1, 0)
K (l3 , m3 , n3) (0, 0, 1)
Rotation of Axes in Three Dimension…
k
i j
x
y
z
I
JK
Z
X
Y
In OXYZ, i= (l1, l2, l3)
O
j = (m1, m2, m3)
k = (n1, n2, n3)
P(x, y, z) or P(X, Y, Z)
r
r = x i+y j+z k
= x (l1I + l2J + l3K) +y (m1I + m2J + m3K) +z (n1I + n2J + n3K)
= (x l1+ ym1+ zn1)I + (x l2+ ym2+ zn2)J +(x l3+ ym3+ zn3)K
Rotation of Axes in Three Dimension…
z
y
x
nml
nml
nml
Z
Y
X
333
222
111
r = x i+y j+z k
= (x l1+ ym1+ zn1)I + (x l2+ ym2+ zn2)J +(x l3+ ym3+ zn3)K
=X I+Y J+Z K
Z
Y
X
nnn
mmm
lll
z
y
x
321
321
321
Rotation of Axes in Three Dimension…
z
y
x
nml
nml
nml
Z
Y
X
333
222
111
Z
Y
X
nnn
mmm
lll
z
y
x
321
321
321
1
321
321
321
333
222
111
nnn
mmm
lll
nml
nml
nml
Plane
x
y
z
r
O
n
a
P(x0 , y0 , z0)Q(x, y, z)
( a - r )· n = 0
r · n = a · n
-- Vector equation of a plane
ax+by+cz=ax0+by0+cz0
or a(x-x0) + b(y-y0) +c(z-z0) =0
If the normal n=(a, b, c), then the equation for the plane can be written as:
QP = a-r
QP · n = 0
Rotation of Axes in 3 Dimensions
x
z
y
X’
Y’
Z’
i
jk
J
IK
^
^ ^
^
^
^
Rotation of Axes in 3 Dimensions
x
z
y
i
jk ^
^
Il1
m1
I = (l1, m1, n1)^
n1
Rotation of Axes in 3 Dimensions
x
z
y
i
jk ^
^
J
l2
m2
n2 J = (l2, m2, n2)^
Rotation of Axes in 3 Dimensions
x
z
y
i
jk ^
^
K
l3
m3
n3
K = (l3, m3, n3)^
Rotation of Axes in 3 Dimensions
x
z
y
X’
Y’
Z’
i
jk
J
IK
^
^ ^
^
^
^l1
l2
l3
i = (l1, l2, l3)
In the X’, Y’, Z’ system
^
Rotation of Axes in 3 Dimensions
x
z
y
X’
Y’
Z’
i
jk
J
IK
^
^ ^
^
^
^
j = (m1, m2, m3)
In the X’, Y’, Z’ system
^
m2
m1
m3
Rotation of Axes in 3 Dimensions
x
z
y
X’
Y’
Z’
i
jk
J
IK
^
^ ^
^
^
^
k = (n1, n2, n3)
In the X’, Y’, Z’ system
^
n1
n2
n3
Rotation of Axes in 3 Dimensions
x
z
y
X’
Y’
Z’
i
jk
J
IK
^
^ ^
^
^
^
P(x, y, z) or P(X’, Y’, Z’)are related by Direction Cosines
Example
Find the equation of a line which passes through P(1, 2, -6) and is parallel to the vector (3, 1, -1)
Example
Find the equation of a plane which passes through P(1, 2, -6) and is perpendicular to the vector (3, 1, -1)