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Dot Product Cross Product of two vectors

Work done by a force

F

W F s cos?

?

F s

s

F

?

s

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

Properties of Dot Product

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

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

(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

Example

- Find the angle between vectors a (1, 1, -1) and

b (2, -1, 0) - a b 1x2 1x(-1) (-1)x0 1
- cos ?

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

Example

- a a 2 , b ß - , c -

? - Find the numbers a, ß, ? which make the vectors

a, b and c mutually perpendicular.

Example

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

Direction Cosines

z

a

?z

?y

?x

y

x

Example

- Find the direction cosines of the vector

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.

Direction Ratios of a straight line L

Line L

, ,

s

p q r

Example

- (Two dimension) Find a set of direction ratios

for the straight line y2x1.

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)

(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

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 ?

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

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

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

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

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

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

Rotation of Axes in Three Dimension

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

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

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)