Question

4. For the vectors b = (1, 2, −2) and a = (−3, 0, 4) (a) Compute the projection of vector b onto the line along vector a as p = ˆxa. (b) Compute the projection of vector b onto the line along vector a as p = Pb. (c) Compute the error vector. (d) Compute the length of the projection vector and length of the error vector.

Answer 1

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

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)

`p=(at*b)/(at*a)*a\\ p=((-3, 0, 4)*(1, 2,-2))/((-3, 0, 4)*(-3, 0, 4)) *(-3, 0, 4)\\ p=(-3+0-8)/(9+0+16)*(-3, 0, 4)\\ p=-(11)/(25) *(-3, 0, 4)\\ p=((33)/(25) ,0,-(44)/(25))`

c)

`e=b-p\\ e=(1, 2, -2)-((33)/(25) ,0,-(44)/(25))\\ e=(-(8)/(25) ,2,-(6)/(25) )`

d) We use the formula

`p=\sqrt{((33)/(25) )^(2) +(0)^(2) +((-44)/(25) )^(2)} =(11)/(5)`

e) Applying the same formula we have

`e=\sqrt{((-8)/(25) )^(2) +(2)^(2) +((-6)/(25) )^(2)} =(2)/(5)√(26) =2.039`

Answer 2

A) (33/5, 0, 44/5)

B) - 11

C) (28/5, 2, - 54/5)

D) 11, 12.83

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