If l12+m12+n12=1 etc., and l1 l2+m1 m2+n1 n2 =0 etc., and Δ=| l1 m1 n1 l2 m2 n2 l3 m3 n3 | then

Question

If l12+m12+n12=1 etc., and l1 l2+m1 m2+n1 n2 =0 etc., and Δ=| l1 m1 n1 l2 m2 n2 l3 m3 n3 | then (A) |Δ|=3 (B) |Δ|=2 (C) |Δ|=1 (D) Δ=0

Tardigrade Admin · Accepted Answer

We have Δ2=| l1 m1 n1 l2 m2 n2 l3 m3 n3 || l1 m1 n1 l2 m2 n2 l3 m3 n3 | =| a1 b21 b31 b12 a2 b32 b13 b23 a3 | text where ak=lk2+mk2+nk2=1 text for k=1,2,3 text and bi j=li lj+mi mj+ni nj=0 ∀ i ≠ j text Thus, Δ2=| 1 0 0 0 1 0 0 0 1 |=1 ⇒|Δ|=1

Q. If $l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = 1$ etc., and $l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$ $= 0$ etc., and $Δ = ∣ ∣ l_{1} l_{2} l_{3} m_{1} m_{2} m_{3} n_{1} n_{2} n_{3} ∣ ∣$ then

$∣Δ∣ = 3$

$∣Δ∣ = 2$

$∣Δ∣ = 1$

$Δ = 0$

Q. If l12​+m12​+n12​=1 etc., and l1​l2​+m1​m2​+n1​n2​ =0 etc., and Δ=∣∣​l1​l2​l3​​m1​m2​m3​​n1​n2​n3​​∣∣​ then

∣Δ∣=3

∣Δ∣=2

∣Δ∣=1

Δ=0

Q. If $l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = 1$ etc., and $l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$ $= 0$ etc., and $Δ = ∣ ∣ l_{1} l_{2} l_{3} m_{1} m_{2} m_{3} n_{1} n_{2} n_{3} ∣ ∣$ then

$∣Δ∣ = 3$

$∣Δ∣ = 2$

$∣Δ∣ = 1$

$Δ = 0$