Question

Let H=Span{v₁,v₂} and K=Span{v₃,v₄}, where v₁,v₂,v₃,v₄ are given below.

v₁ = [3 2 5], v₂ =[4 2 6], v₃ =[5 -1 1], v₄ =[0 -21 -9]

Then H and K are subspaces of R³ . In fact, H and K are planes in R³ through the origin, and they intersect in a line through 0. Find a nonzero vector w that generates that line.

w = { _______ }

Answer 1

By given condition we have, span{v₁,v₂} = span{v₃,v₄}

i.e., av₁+bv₂ = cv₃+dv₄

i.e., a(3,2,5)+b(4,2,6) = c(5,-1,1)+d(0,-21,-9)

i.e., 3a+4b = 5c...........(i)

2a+2b = -c-21d........(ii)

5a+6b = c-9d..........(iii)

From (i) we get, c = (3/5)a+(4/5)b

Putting this in (ii) we get, d = -(13/105)a-(2/15)b

Putting values of c and d in (iii) we get,

5a+6b = (12/7)a+2b

i.e., (23/7)a+4b = 0

i.e., 23a+28b = 0

i.e., a = -(28/23)b

Let us take b = 23k. Then, a = -28k.

Then, a(3,2,5)+b(4,2,6) = -28k*(3,2,5)+23k*(4,2,6)

i.e., a(3,2,5)+b(4,2,6) = k(8,-10,-2)

Therefore, w = (8,-10,-2).