Find all values of a and b so that vectors <-a,2b,a> and <a,b,1> are perpendicular and have same length

Question

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Answer 1

same length implies a^2+4b^2+a^2=a^2+b^2+1
a^2+3b^2=1
Perpendicular implies dot product =0
-a^2+2b^2+a=0
so b^2 =(1-a^2)/3
-a^2+(1-a^2)/3 +a=0
-4a^2+3a+1=0
a=1 and a = -1/4
for a=1 b=0
for a=-1/4
b^2= 5/16 so b=+-1/4*sqrt(5)

Answer 2

-a^2+2b^2+a=0 because the dot product has to be zero

or
a^2-a=b^2