64 Equations
Lines and
planes
I
Line:
Angle
planes
between two
(x,y,z) (x,y,z) + t(a,b,c)
=
point
I Ldirectional
Vector
cos0 b
=
can write in
parametric eq.:
2x.x 1+5t
=
z=4t Matrix
multiplication:
2 3t
y
+ row xcolumn: 1
entry
=
Planes: (Dot product, of normal and some other vector in the
plane) k=7-5.
xx. (1,2,3): 7x -
5,y
-
4,z + 3) 0
=
ND,a
C
normal n =
(1, 2, 3,
minerations:
simplifiesas point <5,4.-37
=
1) R, <> Rc
|(x 5) + 4) 2)2xD, 5R,
-
2(y
-
+ 3(z 3)
+ a
=
or
bz + 9 0 3)R, 5R2
2y
+
=
x -
5 + -
8 +
-
ay 3z=4-cartesian
equation
x+
+
these operations
can
only use row
Systems (Linear equations
sE
130 25 sio'st
No Solution
Unique Solution Infinitely many solutions
Row
Operations i
1)
Interchanging two rows
R. *
Ra
1)
Multiplying a row
by a constant 5Rs
3)
Adding/subtractingfrom
of a row
a
multiple
another
Re + 5R, or B.-fRm
, Matrices:
Inverse: Determinants:
Transform matrix to
upper
A=
A I I 000 identity matrix I find
triangular
=
or lower
1)
product of
diagonals.
Augmented
GcEasoodtre
matrix with I =
2)Transform A to I ⑤8
3) I becomes new matrix (inverse of Al
2x2 matrix: NBd -Matrices are not
commutative
A =
(7 (Al =
ad -
be ABFBA
=> A"bEbAT
=>
A" =
A
Ed]
A only exists if determinant of matrix is not
equal to 0 (IA) =0)
# erties:
·
IAB1 =
1AlIB ·
(Al =
IAT
·
IA) =
kPIA) OR IKA" =
KPIA" ·
Square matrix is invertable if and only
A "1
· =
TA if det (A) = 0
Cramer's Rule
x=
·
Aman
size:
·
Indicated
by mxn (rows x columns).
·
Individual locations /entries matrix
given by aij
within the
-
i -
row
-
j-column
Lines and
planes
I
Line:
Angle
planes
between two
(x,y,z) (x,y,z) + t(a,b,c)
=
point
I Ldirectional
Vector
cos0 b
=
can write in
parametric eq.:
2x.x 1+5t
=
z=4t Matrix
multiplication:
2 3t
y
+ row xcolumn: 1
entry
=
Planes: (Dot product, of normal and some other vector in the
plane) k=7-5.
xx. (1,2,3): 7x -
5,y
-
4,z + 3) 0
=
ND,a
C
normal n =
(1, 2, 3,
minerations:
simplifiesas point <5,4.-37
=
1) R, <> Rc
|(x 5) + 4) 2)2xD, 5R,
-
2(y
-
+ 3(z 3)
+ a
=
or
bz + 9 0 3)R, 5R2
2y
+
=
x -
5 + -
8 +
-
ay 3z=4-cartesian
equation
x+
+
these operations
can
only use row
Systems (Linear equations
sE
130 25 sio'st
No Solution
Unique Solution Infinitely many solutions
Row
Operations i
1)
Interchanging two rows
R. *
Ra
1)
Multiplying a row
by a constant 5Rs
3)
Adding/subtractingfrom
of a row
a
multiple
another
Re + 5R, or B.-fRm
, Matrices:
Inverse: Determinants:
Transform matrix to
upper
A=
A I I 000 identity matrix I find
triangular
=
or lower
1)
product of
diagonals.
Augmented
GcEasoodtre
matrix with I =
2)Transform A to I ⑤8
3) I becomes new matrix (inverse of Al
2x2 matrix: NBd -Matrices are not
commutative
A =
(7 (Al =
ad -
be ABFBA
=> A"bEbAT
=>
A" =
A
Ed]
A only exists if determinant of matrix is not
equal to 0 (IA) =0)
# erties:
·
IAB1 =
1AlIB ·
(Al =
IAT
·
IA) =
kPIA) OR IKA" =
KPIA" ·
Square matrix is invertable if and only
A "1
· =
TA if det (A) = 0
Cramer's Rule
x=
·
Aman
size:
·
Indicated
by mxn (rows x columns).
·
Individual locations /entries matrix
given by aij
within the
-
i -
row
-
j-column
