Respuesta :
Answer:
A) (33/5, 0, 44/5)
B) - 11
C) (28/5, 2, - 54/5)
D) 11, 12.83
Step-by-step explanation:
A)
Given the vectors
b= (1, 2,-2)
a= (-3, 0, 4)
Projection of the vector b onto the line along vector a as p=ˆxa
Calculating ab,
ab= a1b1 + a2b2 + a3b3
a1= - 3, a2=0, a3=4, b1=1, b2=2, b3= - 2
ab= (-3)(1) + (0)(2) +(4)(-2)
ab= - 3 + 0 +(-8)
ab= - 11
Vector projection which is
( ab÷ /vector a/^2) × vector a
= - 11/√(-3)^2 + (0)^2 + (4)^2
= - 11/ √9 +16
=-11/√25
= - 11/5× (-3, 0, - 4)
= (33/5, 0, 44/5)
B) When p= pb
It will be a scalar projection and will be written as:
ab÷ /a/
-11/√1
= - 11
C) Given the vector that form P to b in
e= b - p
=b- ˆxa
e= (1, 2,-2) - (33/5, 0, 44/5)
= (1 - 33/5, 2- 0, -2 - 44/5)
=(5-33/5, 2, - 10- 44/5)
= (28/5, 2, - 54/5)
D.
Length of the projection vector:
/e/ = √(33/5^2 + 0 + 44/5^2)
/e/= √33^2/25 + 0 + 44^2/25
/e/= √33^2/25 +44^2/25
/e/= √121
/e/= 11
Length of error vector
/e/ = √(28/5)^2 + 2^2 + (-54/5)^2
/e/= √28^2/25 +4 +(-54^2/25)
/e/= 12.83
Answer:
a) and b) p= (33/25, 0, -44/25)
c) e = (-8/25, 2, -6/25)
d) p = 11/5 = 2.2
e) e = (2/5)√26 = 2.039
Step-by-step explanation:
Given
b = (1, 2, −2)
a = (−3, 0, 4)
a) and b) We use the formula
p = (at*b)/(at*a)*a
⇒ p = ((−3, 0, 4)*(1, 2, −2)/((−3, 0, 4)*(−3, 0, 4)))*(−3, 0, 4)
[tex]p=\frac{at*b}{at*a}*a\\ p=\frac{(-3, 0, 4)*(1, 2,-2)}{(-3, 0, 4)*(-3, 0, 4)} *(-3, 0, 4)\\ p=\frac{-3+0-8}{9+0+16}*(-3, 0, 4)\\ p=-\frac{11}{25} *(-3, 0, 4)\\ p=(\frac{33}{25} ,0,-\frac{44}{25})[/tex]
c)
[tex]e=b-p\\ e=(1, 2, -2)-(\frac{33}{25} ,0,-\frac{44}{25})\\ e=(-\frac{8}{25} ,2,-\frac{6}{25} )[/tex]
d) We use the formula
[tex]p=\sqrt{(\frac{33}{25} )^{2} +(0)^{2} +(\frac{-44}{25} )^{2}} =\frac{11}{5}[/tex]
e) Applying the same formula we have
[tex]e=\sqrt{(\frac{-8}{25} )^{2} +(2)^{2} +(\frac{-6}{25} )^{2}} =\frac{2}{5}\sqrt{26} =2.039[/tex]
