Express the vector v as the sum of a vector parallel to b and a vector orthogonal to b. v = 4i - 2j + 6k, b= -2i+j-3k

proximumha 2022-08-05 Answered
Express the vector v as the sum of a vector parallel to b and a vector orthogonal to b.
v = 4i - 2j + 6k, b= -2i+j-3k
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Andres Barrett
Answered 2022-08-06 Author has 14 answers
parallel to b means vector =ab for some constant a
u orthogonal to b meansu.b = 0
We need u + ab = (4,-2,6)(i)
u.b = 0 means -2ui + uj -3uk = 0 (ii)
From (i):
(ui,uj,uk) + (4a, -2a,6a) = (4,-2,6)
ui + 4a = 4
uj - 2a = -2
uk +6a = 6
eliminate the u's from (ii) to get an equation in a:
-2(4-4a) + (2a -2) -3(6-6a)=0
-8+8a+2a-2-18+18a=0
28a = -28
a = -1
So
ui = 8
uj = -4
uk = 12
So the parallel vector is:
(2,-1,3)
An orthogonal vector is:
(8,-4,12)
Finally we need to scale the orthogonal vector to give the correctlength, so:
(2,-1,3) + b(8,-4,12) = (4,-2,6)
where b is the scale factor
This is satisfied for b=1/4
Hence the 2 vectors are:
(2,-1,3) and (2,-1,3)
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions