Midpoint formula: M 5

x

1

1 x

2

2

;

y

1

1 y

2

2

Slope formula: m 5

y

2

2 y

1

x

2

2 x

1

Slope and Angle of Inclination

Slope Angle of Inclination

Definition Steepness of

a line

Angle formed in relation to the x-axis

Notation m α

Computation

Y

2

2 Y

1

X

2

2 X

1

α is the angle formed between the lines

Relation m 5 tan α

Parallel lines have equal slopes.

Perpendicular lines have negative reciprocal slopes.

Equations for

Lines: y 5 mx 1 b; y 5 m(x 2 x

1

) 1 y

1

Circles: x

2

1 y

2

5 r

2

(radius r, center the origin); (x 2 h)

2

1 (y 2 k)

2

5 r

2

(radius r,center(h, k))

A tangent to a circl e is a line that touches the circle at only one point.

Theorem: A tangent to a circle is perpendicular to the radius to the point of tangency.

4.11 Vector

Vector equation of a straight line

O

P

A

c

a

r

62 Mathematical Formulas for Industrial and Mechanical Engineering

r

!

5 a

!

1 tc

!

, t: scalar parameter

a

!

: position vector of a fixed point on the straight line

c

!

: direction vector

r

!

: position vector of any point on straight line

r

!

5 a

!

1 tðb

!

2 a

!

Þ

B

P

A

O

b

r

a

The length (magnitude) of the 2D vector a 5,a

1

, a

2

. is given by

jaj5

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

a

2

1

1 a

2

2

q

The length (magnitude) of the 3D vector a 5,a

1

, a

2

, a

3

. is given by

jaj5

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

a

2

1

1 a

2

2

1 a

2

3

q

Given a non-zero vector a,aunit vector u (vector of length one) in the same direction

as the vector a can be constructed by multiplying a by the scalar quantity 1=jaj,thatis,

forming

u 5

1

jaj

a 5

a

jaj

The dot product of two vectors gives a scalar that is computed in the following

manner:

In 2D, if a 5,a

1

, a

2

. and b 5,b

1

, b

2

., then dot product 5 a b 5 a

1

b

1

1 a

2

b

2

In 3D, if a 5,a

1

, a

2

, a

3

. and b 5,b

1

, b

2

, b

3

., then dot product 5 a b 5

a

1

b

1

1 a

2

b

2

1 a

3

b

3

If the dot product 5 0, the vectors are perpendicular

Angle between two vectors: Given two vectors a and b separated by an angle θ,

0 # θ # π.

Then

cos θ 5

a b

jajjbj

63Analytic Geometry and Trigonometry

Solving for θ gives

θ 5 arccos

a b

jajjbj

The cross product (it is defined only for vectors of length 3, i.e., 3D) of two

vectors a and b give s a vector perpendicular to the plane of a and b that is

computed in the following manner:

axb

b

θ

)()()(

122113312332

21

21

31

31

32

32

321

321

321

321

321

321

321

321

babakbabajbabai

bb

aa

k

bb

aa

j

bb

aa

i

bbb

aaa

kji

bbb

aaa

kji

bbb

aaa

kji

bbb

aaa

kji

ba

−+−−−−

+−−

+−−−×

64 Mathematical Formulas for Industrial and Mechanical Engineering

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