Question #310259
  1. For what values of h the vectors


⟶ [ 1 ⟶ [ -1 ⟶ [ 5

u = -3 u2 = 9 u3 = -7 <--- u1, u2, and u3 are vectors

-2] -6] h]


are linearly independent?


1
Expert's answer
2022-03-21T02:39:31-0400

x1=(1,1,5),x2=(3,9,7),x3=(2,6,h)k1x1+k2x2+k3x3=0k1=k2=k3=0k1(1,1,5)+k2(3,9,7)+k3(2,6,h)==(0,0,0)k13k22k3=0k1+9k26k3=05k17k2+hk3=0Δ=13219657h=9h+9014++903h42=6h+1240h1246h623\vec{x_1}=(1,-1,5), \vec{x_2}=(-3,9,-7), \vec{x_3}=(-2,-6,h)\\ k_1\vec{x_1}+k_2\vec{x_2}+k_3\vec{x_3}=\vec{0}\\ k_1=k_2=k_3=0\\ k_1(1,-1,5)+k_2(-3,9,-7)+k_3(-2,-6,h)=\\=(0,0,0)\\ k_1-3k_2-2k_3=0\\ -k_1+9k_2-6k_3=0\\ 5k_1-7k_2+hk_3=0\\ \Delta= \begin{vmatrix} 1& -3&-2 \\ -1 & 9&-6\\ 5&-7&h \end{vmatrix}=9h+90-14+\\+90-3h-42=6h+124\neq0\\ h\neq-\frac{124}{6}\\ h\neq-\frac{62}{3}\\



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