The vector equation of the line of intersection of the planes r⃗ =b⃗ +λ1(b⃗ −a⃗ )+μ1(a⃗ −c⃗ ) and r⃗ =b⃗ +λ2(b⃗ −c⃗ )+μ2(a⃗ +c⃗ )a⃗ ,b⃗ ,c⃗  being non-coplanar vectors, is

[a] At the point of intersection of the two given planes, we have$b⃗+\lambda 1\left(b⃗-a⃗\right)+\mu 1\left(a⃗+c⃗\right)=c⃗+\lambda 2\left(b⃗-c⃗\right)+\mu 2\left(a⃗+b⃗\right)$⇒(−λ+μ1−μ2)a⃗ +(1+λ1−λ2−μ2)b⃗ +(μ1−1+λ2)c⃗ =0⃗ ⇒−λ1+μ1−μ2=0,1+λ2−λ2μ2=0,$\text{and}\mu 1-1+\lambda 2=0$[∴ a⃗ ,b⃗ ,c⃗  are non-coplanar vectors]$Fromthel\ast twoequations,we\ge t$λ1+μ1−μ2=0$onsolv\in gthisequationwith-\lambda 1+\mu 1-\mu 2=0,we\ge t$μ1=μ2 and λ1=0$\therefore \lambda 1=0,\mu 1=\mu 2\text{and}\lambda 2=1-\mu 1$on substituting these values in either of the given equations, we obtains r⃗ =b⃗ +μ1(a⃗ +c⃗ )$Astherequiredl\in eof\int er\sec tionofthegivenpla\ne s.$