Dot Product

1
Dot Product Cross Product of two vectors
2
Work done by a force
F
W F s cos?
?
F s
s
F
?
s
3
Dot product (Scalar product)
c
b

• a b a b cos?
• axbx ayby azbz
• 0o lt?lt180o 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 a2

?
d
a
4
Properties of Dot Product

• Commutative property
• a b ba
• Distributive property
• a ( b c ) a b bc

5
Example

• a (1, 2, 4), b (-1, 2, -1)
• a b 1x(-1) 2x2 4x(-1) -1

6
Example

• a (0, 1, -1), b (2, -1, 1)
• a b 0x2 1x(-1) (-1)x1 -2

7
Example
(1,0,0) (1,0,0) 1




(0,1,0) (0,1,0) 1
z


(0,0,1) (0,0,1) 1
1


(1,0,0) (0,1,0) 0
1


(0,1,0) (0,0,1) 0
y
1


(1,0,0) (0,0,1) 0
x
8
Example

• Find the angle between vectors a (1, 1, -1) and
  b (2, -1, 0)
• a b 1x2 1x(-1) (-1)x0 1
• cos ?

9
Example

• A(2,1, 0), B(1, -1,1), C(0, 2, 1) are three
  points. Find the angles in the triangle ABC

B
ß
a
?
A
C
10
Example

• a a 2 , b ß - , c -
  ?
• Find the numbers a, ß, ? which make the vectors
  a, b and c mutually perpendicular.

11
Example

• a 2 , b -
• Construct any vector perpendicular to a and b

12
Direction Cosines
z
a
?z
?y
?x
y
x
13
Example

• Find the direction cosines of the vector

14
Example

• Find the unit vector in the direction of the
  vector a(3, 4, 1).

15
Direction Ratios of a straight line

• To determine the inclination of a straight line.
• Components of any vector s that is parallel to
  line.

Direction Ratios of a straight line L
Line L
, ,
s
p q r
16
Example

• (Two dimension) Find a set of direction ratios
  for the straight line y2x1.

17
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).

18
Components of a vector a(ax, ay, az)
(ax, ay, az)(1, 0,0)ax
z
(ax, ay, az)(0, 1,0)ay
a
1
(ax, ay, az)(0, 0,1)az
1
y
1
x
19
Rotation of Axes in Two dimensions
(cos?, sin?)
y
Y
(cos(p/2?), sin(p/2 ?) (-sin ?, cos ?)
P(x, y), P(X, Y)
X
X
(x, y)(cos?, sin ?) xcos ? ysin ?
?
x
Y
(x, y)(-sin?, cos ?) -xsin ? ycos ?
20
Rotation of Axes in Three Dimension
Z
z
a(x, y, z) x iy jzk in Oxyz
a (?, ?, ?) in OXYZ
a
K
J
Y
y
O
I
x
X
21
Rotation of Axes in Three Dimension
Z
z
In OXYZ, I(1, 0, 0)
J(0, 1, 0) K(0, 0, 1)
K
J
Y
n1
y
In Oxyz, I (l1, m1, n1)
O
m1
l1
I
J (l2, m2, n2) K (l3, m3, n3)
x
X
22
Rotation of Axes in Three Dimension
Z
z
In xyz, i(1, 0, 0)
j(0, 1, 0) k(0, 0, 1)
K
J
Y
l3
y
In OXYZ, i (l1, l2, l3)
O
l2
I
j (m1, m2, m3) k (n1, n2, n3)
l1
x
X
23
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)
24
Rotation of Axes in Three Dimension
Z
In OXYZ, i (l1, l2, l3)
P(x, y, z) or P(X, Y, Z)
z
j (m1, m2, m3) k (n1, n2, n3)
r
K
J
Y
y
O
r x iy jz k x (l1I l2J l3K) y (m1I
m2J m3K) z (n1I n2J n3K)
I
(x l1 ym1 zn1)I (x l2 ym2 zn2)J (x l3
ym3 zn3)K
x
X
25
Rotation of Axes in Three Dimension
r x iy jz k
(x l1 ym1 zn1)I (x l2 ym2 zn2)J (x l3
ym3 zn3)K
X IY JZ K
26
Rotation of Axes in Three Dimension
27
Plane
z
Q(x, y, z)
P(x0 , y0 , z0)
n
( a - r ) n 0
r
a
r n a n
y
O
-- Vector equation of a plane
If the normal n(a, b, c), then the equation for
the plane can be written as
x
axbyczax0by0cz0 or a(x-x0) b(y-y0)
c(z-z0) 0
28
Rotation of Axes in 3 Dimensions
29
Rotation of Axes in 3 Dimensions
30
Rotation of Axes in 3 Dimensions
31
Rotation of Axes in 3 Dimensions
32
Rotation of Axes in 3 Dimensions
33
Rotation of Axes in 3 Dimensions
34
Rotation of Axes in 3 Dimensions
35
Rotation of Axes in 3 Dimensions
P(x, y, z) or P(X, Y, Z) are related by
Direction Cosines
36
Example

• Find the equation of a line which passes through
  P(1, 2, -6) and is parallel to the vector (3, 1,
  -1)

37
Example

• Find the equation of a plane which passes through
  P(1, 2, -6) and is perpendicular to the vector
  (3, 1, -1)