Examples Of Regular Matrix, Singular Matrix, Idempotent Matrix, Involutive Matrix And Orthogonal Matrix

Classified in Physics

Written at September 3, 2020 on English with a size of 9.24 KB.

VECTOR DETERMINE BY LENGTH AND ANGLE
V=<||V|| Cosθ,||V|| Sinθ> ---> ||V||Cosθi + ||V||Sinθj
Ex: a) Find the vector of length 2 that makes an angle of pi/4 with the +axis
b) Find the angle that the vector V=- $\sqrt{\ }$ ₃ i + j makes with the + xaxis
a)<||V||Cosθ , ||V||Sinθ> = <2cos45,2sin45> ---> < $\sqrt{\ }$ ₂, $\sqrt{\ }$ ₂>
b) Normalize... ||V||= $\sqrt{\ }$ _{(-3)² + 1²} = $\sqrt{\ }$ ₄= 2 -----> V/||V||=<- $\sqrt{\ }$ _3/2 , 1/2> =<cosθ,sinθ> ----> cosθ= - $\sqrt{\ }$ _3/2, sinθ=1/2 ---> θ=5pi/6
DOT PRODUCT
If U=<U₁,U₂> and V=<V₁,V₂> then, the dot product is UV+U₁V₁+U₂V₂Ex: a) U=<3,5>, V=<-1,2> -----> UV= (3)(-1)+(5)(2) ---> UV= -3+7 --> UV=7
b) U=<1,-3,4>, V=<1,5,2> ----> UV=(1)(1)+(5)(-3)+(2)(4) ---> UV=-6
ANGLES BETWEEN VECTORS
Cosθ=UV / ||U||||V||
Ex: Find the angle between the vector U= i - 2j + k and a)V=-3i+6j+2k
U=<1,-2,1>; V=<-3,6,2>   ----> ||U||= $\sqrt{\ }$ _{1² + (-2)² +1²} = $\sqrt{\ }$ ₆
||V||= $\sqrt{\ }$ _{(-3)² +6²+2²}= $\sqrt{\ }$ 49 = 7   ---> UV= 1(-3)+(-2)(6)+(1)(2)=-13
Cosθ=UV/||U||||V|| = -13/7 $\sqrt{\ }$ ₆ ; θ=Cos^-1(-13/7 $\sqrt{\ }$ ₆)
DECOMPOSING VECTORS IN ORTHOGONAL COMPONENTS
V=<V₁,V₂> = <||V||Cosθ,||V||Sinθ>
e₁=<1,0> e₂=<1,0>
UXe₁=V₁1+V₂0= V₁   UXe₂= V₂0+V₂1=V₂U=V₁e₁ +V₂e₂U=(VXe₁)e₁+(VXe₂)e₂Ex: V=<2,3> ; e₁:<1/ $\sqrt{\ }$ ₂ , 1/ $\sqrt{\ }$ ₂> , e₂:<-1/ $\sqrt{\ }$ ₂ , 1/ $\sqrt{\ }$ ₂>
VXe₁= 2 X 1/ $\sqrt{\ }$ ₂ + 3 X 1/ $\sqrt{\ }$ ₂= 5/ $\sqrt{\ }$ ₂VXe₂=2 X -1/ $\sqrt{\ }$ ₂ + 3 X 1/ $\sqrt{\ }$ ₂= 1/ $\sqrt{\ }$ ₂V=5/ $\sqrt{\ }$ ₂e₁+1/ $\sqrt{\ }$ ₂e₂ -----> 5/ $\sqrt{\ }$ ₂ X e₁=5 $\sqrt{\ }$ ₂<1/ $\sqrt{\ }$ ₂,1/ $\sqrt{\ }$ ₂> = <5/2,5/2>
1/ $\sqrt{\ }$ ₂X e₂ = 1/ $\sqrt{\ }$ ₂ <-1/ $\sqrt{\ }$ ₂,1/ $\sqrt{\ }$ ₂> =<-1/2, 1/2>
WORK: Force X distance : ||F||d     W:||FCosθ||||PQ||
CROSS PRODUCT: Matrix 3x3 and 2x2 (la primera fila los nums van afuera) (si hay dos rows con dos columns iguales es = 0 y cambias y multiplicas por -1) (I AND K matrix +, J -)
GEOMETRIC PROPERTIES OF CROSS PRODUCT
U(U X V) = 0     V(U X V) = 0
SCALAR TRIPLE PRODUCT
Volume of parallelepiped: V=|U(V X W)| (lo mismo que cross product)
ALGEBRAIC PROPERTIES OF S.T.P
Lo mismo que s.T.P pero tienes que ver si es igual a 0 (lie in the same plane)
PARAMETRIC EQUATIONS OF LINES
Ex: Find the parametric eq of the line passing through point (1,2,-3) and parallel to V= 4i+5j-7k.... X=1+4t, y=2+5j, z=-3-7t.
Ex: Let L₁ and L₂be the lines
L₁: x=1+4t , y=5-4t, z=-1+5t     L₂: x=2+8t, y=4-3t, z=5+t
a) are the lines parallel. B) do the lines intersect?
a)L₁is parallel to the vector V=<4,-4,5>
L₂is parallel to the vector V=<8,-3,1> (not parallel bc neither is a scalar multiply of the other.
b) 1+4t₁=2+8t₂
    5-4t₁=4-3t₂-1+5t₁=5+t₂(eliminas los 4t₁)= 6= 6+5t₂ ---> 5t₂=0, t₂=0
(lo pones en la 1era eq)= 1+4t₁=2 ---> 4t₁=1 ---> t₁=1/4
(lo pones en la 3era eq) vez si es igual
LINES SEGMENTS
Ex: Find parametric eq describing the line segment joining the points P₁(2,4,-1) and P₂(5,0,7)
V=P₁P₂=<3,-4,8> ---->(Sacas xyt mult P₁ con P₁P_{2 le pones t}) x=2+3t, y=4-4t, z=-1+8t

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