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

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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+U2V2
Ex: 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=V2
U=V1e1 +V2e2         U=(VXe1)e1+(VXe2)e2
Ex: 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{\ }2
VXe2=2 X -1/\sqrt{\ }2 + 3 X 1/\sqrt{\ }2 = 1/\sqrt{\ }2
V=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{\ }2 X 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 L2 be 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)L1 is parallel to the vector V=<4,-4,5>
L2 is 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 P1P2 le pones t) x=2+3t, y=4-4t, z=-1+8t

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