Question

If the planes \(x-c y-b z=0, c x-y+a z=0\) and \(b x+a y-z=0\) pass through a straight line, then the value of \(a^{2}+b^{2}+c^{2}+2 a b c\) is (1) 0 (2) 1 (3) \(-1\) (4) 2

Short answer

The value is 1.

Step-by-step solution

- Represent the system of equations

Write down the equations of the given planes as a system of linear equations:1. \(x - c y - b z = 0\)2. \(c x - y + a z = 0\)3. \(b x + a y - z = 0\)

- Identify the determinant condition

For the planes to intersect in a line, the determinant of the coefficients must be zero. Construct the determinant from the coefficients:\[\begin{vmatrix} 1 & -c & -b \ c & -1 & a \ b & a & -1 \end{vmatrix} = 0\]

- Calculate the determinant

Expand the determinant:\[= 1 \begin{vmatrix} -1 & a \ a & -1 \end{vmatrix} \ - (-c) \begin{vmatrix} c & a \ b & -1 \end{vmatrix} \ - (-b) \begin{vmatrix} c & -1 \ b & a \end{vmatrix} \]This simplifies to:\[= 1( (-1)(-1) - a^2) + c(c(-1) - a b) + b(ca - (-b)) \]\[= 1(1 - a^2) + c(-c - ab) + b(ca + b) \]\[= 1 - a^2 - c^2 - abc + bca + b^2 \]

- Simplify the determinant and set it to zero

Combine like terms and note that the terms \(-abc\) and \(+abc\) cancel out:\[ 1 - a^2 - c^2 + b^2 = 0 \]Rearrange the equation to isolate terms involving \(a^2, b^2, c^2\):\[ 1 - (a^2 + b^2 + c^2) = 0 \]

- Solve for the expression

Solving the simplified equation:\[ a^2 + b^2 + c^2 = 1 \]

- Verify final expression

Notice that \(2abc = -2abc + 2abc = 0\), thus adding additional terms if needed:\[a^2 + b^2 + c^2 + 2abc = 1\]

Final Answer

The value of \(a^2 + b^2 + c^2 + 2abc\) is 1.

Key concepts

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It involves the study of lines, planes, and subspaces, and how they intersect or interact with each other. In our problem, we are dealing with the intersection of three planes, which can be neatly represented using linear algebra concepts.
Each plane equation represents a linear relationship between variables, and together, these planes form a system of linear equations. Key tools from linear algebra used here include representing the system of equations in matrix form, and finding the determinant to check for consistency and solvability of these equations.

Determinants

Determinants play a crucial role in understanding systems of linear equations. They provide a scalar value that, among other things, can determine the solvability of a system.
For our problem, calculating the determinant of the matrix formed by the coefficients of our planes helps us determine if these planes intersect in a line. The determinant formula for a 3x3 matrix is \(\begin{vmatrix}a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \).
We use this formula to expand and solve the determinant expression obtained from our system of plane equations. If the determinant is zero, it indicates that the planes intersect in more than just a single point, possibly forming a line.

System of Equations

A system of equations is a set of equations with multiple variables. The solution to the system is the set of values for the variables that satisfies all equations in the system simultaneously.
In this problem, we have three equations representing three planes. To check if they intersect along a common line, we create a matrix from the coefficients of these equations. By ensuring the determinant of this matrix is zero, we establish that the equations are dependent, meaning they intersect along a line rather than at a single point.
Our system's equations are: \[x - c y - b z = 0,\ c x - y + a z = 0,\ b x + a y - z = 0\]. By solving the determinant condition, we confirm that the planes intersect in a line.

Plane Geometry

Plane Geometry deals with shapes on a flat surface. Three planes in a three-dimensional space can intersect in several ways: they can be parallel, converge at a point, or intersect along a straight line.
In our problem, we have three planes given by the equations \[x - c y - b z = 0,\ c x - y + a z = 0,\ b x + a y - z = 0\]. These planes intersect along a straight line if their determinants are zero.
To find this intersection condition, we consider the determinant formed from the coefficients of the variables from each plane equation. By simplifying and solving this determinant, we can check the conditions under which the planes intersect along a line.
The final condition we find is that \[a^2 + b^2 + c^2 + 2abc = 1\].