Chapter 4.2, Problem 40E

[M] Let H = Span {v₁, v₂} and K = Span {v₃, v₄}, where

${\text{v}}_1=\left[\begin{array}{c}5\\ 3\\ 8\end{array}\right]$, ${\text{v}}_2=\left[\begin{array}{c}1\\ 3\\ 4\end{array}\right]$, ${\text{v}}_3=\left[\begin{array}{c}2\\ -1\\ 5\end{array}\right]$, ${\text{v}}_4=\left[\begin{array}{c}0\\ -12\\ -28\end{array}\right]$

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. [Hint: w can be written as c₁v₁ + c₂v₂ and also as c₃v₃ + c₄v₄. To build w, solve the equation c₁v₁ + c₂v₂ = c₃v₃ + c₄v₄ for the unknown c_j’s.]