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WTW164 Summary

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Document contains all the key themes/topics that appear in WTW164 in a summarized way. Great document to skim through just before writing a test or exam.

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  • November 27, 2022
  • 7
  • 2022/2023
  • Summary
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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

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