If (2+a+b) i+(a+2b+c) j-(b+c) k, (1+b) i+2b j-b k, (2+b) i+2b j+(1-b) k are coplanar, whic…

Question

If $(2+a+b)\hat i+(a+2b+c)\hat j-(b+c)\hat k$, $(1+b)\hat i+2b\hat j-b\hat k$, $(2+b)\hat i+2b\hat j+(1-b)\hat k$ are coplanar, which holds?

Accepted Answer

Set up the determinant of the three vectors and simplify with row operations $R_3\to R_3-R_2$, $R_1\to R_1-R_2$:
$$\begin{vmatrix}a+1&a+c&-c\b+1&2b&-b\1&0&1\end{vmatrix}=0$$

Expanding: $(a+1)(2b)-(a+c)(2b+1)-c(-2b) = 2ab+2b-2ab-a-2bc-c+2bc = 2b-a-c=0$

$2b=a+c$

Solutions · 1