Q) Let $P_{1},P_{2},$ and $P_{3}$ denote, respectively, the planes defined by
$a_{1}x + b_{1}y + c_{1}z =\alpha_{1}$
$a_{2}x + b_{2}y + c_{2}z = \alpha _{2}$
$a_{3}x + b_{3}y + c_{3}z = \alpha _{3}$
It is given that $P_{1},P_{2},$ and $P_{3}$ intersect exactly at one point when $\alpha _{1}= \alpha _{2}= \alpha _{3}=1$ . If now $\alpha _{1}=2, \alpha _{2}=3 \;and \; \alpha _{3}=4$ then the planes
(A) do not have any common point of intersection
(B) intersect at a unique point
(C) intersect along a straight line
(D) intersect along a plane
My Approach :- Since, System of equations $Ax=b$ has a solution (either unique or infinitely many) when vector b is in the column space of $A$. So, for given vector $b=(1,1,1)$ , $Ax=b$ gives unique solution it means vector $b$ is in the column space of $A$ and all $3$ vectors of $A$ i.e. $(a_{1},a_{2},a_{3}),(b_{1},b_{2},b_{3}),(c_{1},c_{2},c_{3})$ are linearly independent and So, matrix formed by these $3$ vectors is invertible.
Now, if vector $b$ changed into $(2,3,4)$ then how to sure that this vector will be in the same column space. If it is in the same column space then it will give unique solution because all the vectors of $A$ will still be linearly independent and if it is not in the column space of $A$ then there will be no solution , So these $3$ planes neither cut at a point nor a straight line(infinitely many solution). So, I am confused with option $(A)$ and $(B)$. Please help.
Best Answer
The key thing you pointed out is that $Ax = b$ having a unique solution for some $b$ shows that the columns of $A$ are linearly independent. In particular, the columns of $A$ are three linearly independent vectors in $\mathbb{R}^3$, so they actually form a full basis for $\mathbb{R}^3$. By the definition of a basis, every vector in $\mathbb{R}^3$, such as the vector $(2,3,4)$, can be uniquely written as a linear combination of $A$'s columns. This unique linear combination corresponds to the unique solution to $Ax = (2,3,4)$, so the answer is (B).