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Analytic geometry
Analytic geometry
10 formulas · Mathematics
Distance between two points
∣
A
B
∣
=
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
|AB|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
∣
A
B
∣
=
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
Midpoint of a segment
S
=
(
x
1
+
x
2
2
,
y
1
+
y
2
2
)
S=\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)
S
=
(
2
x
1
+
x
2
,
2
y
1
+
y
2
)
Slope-intercept equation of a line
y
=
a
x
+
b
y=ax+b
y
=
a
x
+
b
Slope from two points
a
=
y
2
−
y
1
x
2
−
x
1
a=\dfrac{y_2-y_1}{x_2-x_1}
a
=
x
2
−
x
1
y
2
−
y
1
Condition for parallel lines
a
1
=
a
2
a_1=a_2
a
1
=
a
2
Condition for perpendicular lines
a
1
⋅
a
2
=
−
1
a_1\cdot a_2=-1
a
1
⋅
a
2
=
−
1
Distance from a point to a line
d
=
∣
A
x
0
+
B
y
0
+
C
∣
A
2
+
B
2
d=\dfrac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}
d
=
A
2
+
B
2
∣
A
x
0
+
B
y
0
+
C
∣
Equation of a circle
(
x
−
a
)
2
+
(
y
−
b
)
2
=
r
2
(x-a)^2+(y-b)^2=r^2
(
x
−
a
)
2
+
(
y
−
b
)
2
=
r
2
Magnitude of a vector
∣
u
⃗
∣
=
x
2
+
y
2
|\vec{u}|=\sqrt{x^2+y^2}
∣
u
∣
=
x
2
+
y
2
Dot product
u
⃗
∘
v
⃗
=
x
1
x
2
+
y
1
y
2
=
∣
u
⃗
∣
∣
v
⃗
∣
cos
α
\vec{u}\circ\vec{v}=x_1x_2+y_1y_2=|\vec u||\vec v|\cos\alpha
u
∘
v
=
x
1
x
2
+
y
1
y
2
=
∣
u
∣∣
v
∣
cos
α
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