# Determine for all values of the parameter p whether the statement d in <a,b,c> holds for the vectors a,b,c,d in RR^4: a=3,−1,2,1)^T , b=(15,8,8,7)^T , c=(12,6,7,p)^T , d=(6,8,−9,12)^T

Determine for all values of the parameter p whether the statement $d\in $ holds for the vectors $a,b,c,d\in {R}^{4}:$ $a=3,-1,2,1{\right)}^{T}$ , $b=\left(15,8,8,7{\right)}^{T}$ , $c=\left(12,6,7,p{\right)}^{T}$ , $d=\left(6,8,-9,12{\right)}^{T}$
d is a subspace of a,b,c ? How can we find all values of the parameter p?
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Dakota Duarte
Hint: first verify that a $4×3$ matrix whose columns are a,b,c is non-singular. Then define another matrix ,A, whose columns are $a,b,c,d$ and find values of p for which $|A|=0$