Question

Consider three planes $$\begin{array}{rl}& P_1:x-y+z=1\\ & P_2:x+y-z=-1\\ & P_3:x-3y+3z=2\end{array}$$ Let $L_1,L_2,L_3$ be the lines of intersection of the planes $P_2$ and $P_3,P_3$ and $P_1$, and $P_1$ and $P_2$, respectively. STATEMENT-1: At least two of the lines $L_1,L_2$ and $L_3$ are non- parallel. and STATEMENT-2: The three planes do not have a common point. (A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

Short answer

(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

Step-by-step solution

Analyze Line Intersections

To find the intersection lines of the planes, we should pair up the plane equations and solve them together. For every pair, we would normally end up with a system of equations whose solution would be a line (if the planes are not parallel and do not coincide).

Examine the Plane Coefficients

By looking at the normal vectors of the planes, given by the coefficients of x, y, and z in each equation, we determine the relationship between the planes. Normal vectors are: for $P_1(1,-1,1)$, for $P_2(1,1,-1)$, and for $P_3(1,-3,3)$. Observing the normal vectors, we can see that no two are proportional, hence none of the planes are parallel. This means that the lines of intersection, $L_1,L_2,L_3$, must exist and at least two lines must intersect or be non-parallel.

Determine if A Common Point Exists

To find out if there is a common point of intersection for all three planes, we can try to solve the system of all three equations simultaneously. If there is a unique solution, then a common point exists. If the system is inconsistent or has no solution, then a common point does not exist.

Solve the System of Equations

To solve, we can use methods such as substitution, elimination or matrix operations like reduced row echelon form. However, a quicker way could be to notice that if we add $P_1$ and $P_2$ we get $2x=0$, and if we add $P_1$ and $P_3$ we get $2x+2z=3$. Since we can clearly solve for $x$ and $z$ from these two new equations, and then together with one of the original equations solve for $y$, we can confirm there is indeed a unique solution to the system. Therefore, the three planes intersect at a common point.

Key concepts

Line of Intersection

When two planes in three-dimensional space intersect, they typically do so along a line. This line of intersection is fundamental in understanding the spatial relationship between planes. To find the intersection line between any two planes, we consider their equations and solve them simultaneously.

For instance, to find the line of intersection between planes P₁ and P₂, we would solve the system comprised of their respective equations. This system of equations usually yields a set of parametric equations or a vector equation for the line. This process is mirrored to find lines L₁, L₂, and L₃, each representing the line of intersection for a different pair of planes from the problem stated.

Normal Vectors

A normal vector of a plane is a three-dimensional vector that is perpendicular to the plane. Each component of the normal vector corresponds to the coefficients of the variables x, y, and z in the plane's equation. In our exercise, the planes P₁, P₂, and P₃ have normal vectors (1, -1, 1), (1, 1, -1), and (1, -3, 3), respectively.

Comparing these vectors is a quick method to determine the relationship between their corresponding planes. If two normal vectors are proportional (scalar multiples of each other), their planes are parallel. In our case, none of the vectors are proportional, indicating that none of the planes are parallel and thereby confirming the existence of the lines of intersection.

System of Equations

A system of equations consists of multiple equations that we solve together to find a common solution. In the context of plane intersections, a system can be formed by the equations of two or more planes to determine their lines of intersection or a common point. The system is solved using methods such as substitution, elimination, or matrix operations, such as row reduction to echelon form.

In our problem, we are dealing with a system formed by the given equations of three planes. When these planes are not parallel, the system will have solutions that can indicate a line of intersection or a single point where all three planes meet.

Unique Solution

A unique solution to a system of equations means that there is exactly one set of values for the variables that satisfies all the equations. When we have three planes, as in our exercise, a unique solution indicates that the planes intersect at a single point, which is common to all of them.

In this case, the unique solution can be found by various methods. For instance, by adding equations of P₁ and P₂, we simplify the system to find the value for x, and similarly we find z. After that, we can easily determine the value for y, proving that there is indeed one point in space where all three planes intersect, thus confirming the existence of a unique solution to the system.