Question

Known vectors:

a = (α, 0, 1), b = (1, β, 1), c = (1, 1, γ)

Determine the values of α, β,γ when the three vectors are orthogonal to each other.

Answer 1

$a = (α, 0, 1)=αi+k \\ b = (1, β, 1)=i+βj+k \\ c = (1, 1, γ)=i+j+γk\\$

Since a,b, and care pairwise orthogonal to each other

$\bar{a}*\bar{b}=0 \space \& \space \bar{a}*\bar{c}=0\\ (αi + k)(i+βj+k)=0\\ α+0+1=0\\ α=-1$

$(αi + k)(i+j+γk)=0\\ α+0+γ=0\\ α=-γ\\ γ=1$

$(i + βj+k)(i+j+γk)=0\\ 1+β+γ=0\\ 1+β+γ=0\\ β=-2$