Dot Product & Cross Product of two vectors

37
Dot Product & Cross Product of two vectors

description

Dot Product & Cross Product of two vectors. Work done by a force. F. W = F s cos θ. θ. = F · s. s. F. θ. s. Dot product (Scalar product). c. b. a · b = |a| |b| cos θ = a x b x + a y b y + a z b z 0 o < θ

Transcript of Dot Product & Cross Product of two vectors

Page 1: Dot Product & Cross Product of two vectors

Dot Product & Cross Product of two vectors

Page 2: Dot Product & Cross Product of two vectors

Work done by a force

F

s

θ

θW = F s cosθ

= F · s

F

s

Page 3: Dot Product & Cross Product of two vectors

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

Page 4: Dot Product & Cross Product of two vectors

Properties of Dot Product

Commutative property

a ·b = b·a

Distributive property

a · ( b + c ) = a ·b + b·c

Page 5: Dot Product & Cross Product of two vectors

Example

a = (1, 2, 4), b =(-1, 2, -1)

a · b = 1x(-1) + 2x2 + 4x(-1) = -1

Page 6: Dot Product & Cross Product of two vectors

Example

a = (0, 1, -1), b = (2, -1, 1)

a · b = 0x2 + 1x(-1) +(-1)x1 = -2

Page 7: Dot Product & Cross Product of two vectors

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

Page 8: Dot Product & Cross Product of two vectors

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

Page 9: Dot Product & Cross Product of two vectors

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

Page 10: Dot Product & Cross Product of two vectors

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

Page 11: Dot Product & Cross Product of two vectors

Example

a = + +2 , b= + - Construct any vector perpendicular to a and b

i j k i j k

Page 12: Dot Product & Cross Product of two vectors

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

Page 13: Dot Product & Cross Product of two vectors

Example

Find the direction cosines of the vector

k2-j2is ˆˆˆ

Page 14: Dot Product & Cross Product of two vectors

Example

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

Page 15: Dot Product & Cross Product of two vectors

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

Page 16: Dot Product & Cross Product of two vectors

Example

(Two dimension) Find a set of direction ratios for the straight line y=2x+1.

Page 17: Dot Product & Cross Product of two vectors

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).

Page 18: Dot Product & Cross Product of two vectors

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

Page 19: Dot Product & Cross Product of two vectors

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

Page 20: Dot Product & Cross Product of two vectors

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

Page 21: Dot Product & Cross Product of two vectors

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

Page 22: Dot Product & Cross Product of two vectors

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

Page 23: Dot Product & Cross Product of two vectors

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)

Page 24: Dot Product & Cross Product of two vectors

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

Page 25: Dot Product & Cross Product of two vectors

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

Page 26: Dot Product & Cross Product of two vectors

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

Page 27: Dot Product & Cross Product of two vectors

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

Page 28: Dot Product & Cross Product of two vectors

Rotation of Axes in 3 Dimensions

x

z

y

X’

Y’

Z’

i

jk

J

IK

^

^ ^

^

^

^

Page 29: Dot Product & Cross Product of two vectors

Rotation of Axes in 3 Dimensions

x

z

y

i

jk ^

^

Il1

m1

I = (l1, m1, n1)^

n1

Page 30: Dot Product & Cross Product of two vectors

Rotation of Axes in 3 Dimensions

x

z

y

i

jk ^

^

J

l2

m2

n2 J = (l2, m2, n2)^

Page 31: Dot Product & Cross Product of two vectors

Rotation of Axes in 3 Dimensions

x

z

y

i

jk ^

^

K

l3

m3

n3

K = (l3, m3, n3)^

Page 32: Dot Product & Cross Product of two vectors

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

^

Page 33: Dot Product & Cross Product of two vectors

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

Page 34: Dot Product & Cross Product of two vectors

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

Page 35: Dot Product & Cross Product of two vectors

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

Page 36: Dot Product & Cross Product of two vectors

Example

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

Page 37: Dot Product & Cross Product of two vectors

Example

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