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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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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