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Question: If the origin and point \(P\left( {2,3,4} \right),Q\left( {1,2,3} \right)\) and \(R\left( {x,y,z} \r...

If the origin and point P(2,3,4),Q(1,2,3)P\left( {2,3,4} \right),Q\left( {1,2,3} \right) and R(x,y,z)R\left( {x,y,z} \right) are coplanar then
A) x−2y−z=0x - 2y - z = 0
B) x+2y+z=0x + 2y + z = 0
C) x−2y+z=0x - 2y + z = 0
D) 2x−2y+z=02x - 2y + z = 0

Explanation

Solution

It is given in the question that the origin and point P(2,3,4),Q(1,2,3)P\left( {2,3,4} \right),Q\left( {1,2,3} \right) and R(x,y,z)R\left( {x,y,z} \right) are coplanar then what is the equation:
Firstly, we will assume O(0,0,0)O\left( {0,0,0} \right) be the origin of the plane containing point P, Q and R.
Then after, if the points are coplanar then the equation satisfy is \left| {\begin{array}{*{20}{l}} {{x_4} - {x_1}}&{{y_4} - {y_1}}&{{z_4} - {z_1}} \\\ {{x_4} - {x_2}}&{{y_4} - {y_2}}&{{z_4} - {z_2}} \\\ {{x_4} - {x_3}}&{{y_4} - {y_3}}&{{z_4} - {z_3}} \end{array}} \right| = 0 .
Next, we will put the values of P, Q, R and O in the above equation, and Finally after solving the determinant we will get our answer.

Complete step by step solution:
It is given in the question that the origin and point P(2,3,4),Q(1,2,3)P\left( {2,3,4} \right),Q\left( {1,2,3} \right) and R(x,y,z)R\left( {x,y,z} \right) are coplanar then what is the equation:
Let O(0,0,0)O\left( {0,0,0} \right) be the origin of the plane containing point P, Q and R.
Thus, O, P, Q and R are collinear points.
We know that, if A(x1,y1,z1),B(x2,y2,z2),C(x3,y3,z3)A\left( {{x_1},{y_1},{z_1}} \right),B\left( {{x_2},{y_2},{z_2}} \right),C\left( {{x_3},{y_3},{z_3}} \right) and D(x4,y4,z4)D\left( {{x_4},{y_4},{z_4}} \right) are four points such that they are coplanar, then they satisfy the following condition:
\left| {\begin{array}{*{20}{l}} {{x_4} - {x_1}}&{{y_4} - {y_1}}&{{z_4} - {z_1}} \\\ {{x_4} - {x_2}}&{{y_4} - {y_2}}&{{z_4} - {z_2}} \\\ {{x_4} - {x_3}}&{{y_4} - {y_3}}&{{z_4} - {z_3}} \end{array}} \right| = 0
Now, consider the points O(0,0,0),O\left( {0,0,0} \right), P(2,3,4),Q(1,2,3)P\left( {2,3,4} \right),Q\left( {1,2,3} \right) and R(x,y,z)R\left( {x,y,z} \right)
Let,
(x1,y1,z1)=(2,3,4) (x2,y2,z2)=(1,2,3) (x3,y3,z3)=(x,y,z) (x4,y4,z4)=(0,0,0)  \left( {{x_1},{y_1},{z_1}} \right) = \left( {2,3,4} \right) \\\ \left( {{x_2},{y_2},{z_2}} \right) = \left( {1,2,3} \right) \\\ \left( {{x_3},{y_3},{z_3}} \right) = \left( {x,y,z} \right) \\\ \left( {{x_4},{y_4},{z_4}} \right) = \left( {0,0,0} \right) \\\
Now, put the value of (x1,y1,z1),(x2,y2,z2),(x3,y3,z3),(x4,y4,z4)\left( {{x_1},{y_1},{z_1}} \right),\left( {{x_2},{y_2},{z_2}} \right),\left( {{x_3},{y_3},{z_3}} \right),\left( {{x_4},{y_4},{z_4}} \right) in above equation.
Then the equation \left| {\begin{array}{*{20}{l}} {{x_4} - {x_1}}&{{y_4} - {y_1}}&{{z_4} - {z_1}} \\\ {{x_4} - {x_2}}&{{y_4} - {y_2}}&{{z_4} - {z_2}} \\\ {{x_4} - {x_3}}&{{y_4} - {y_3}}&{{z_4} - {z_3}} \end{array}} \right| = 0 become \left| {\begin{array}{*{20}{l}} {0 - 2}&{0 - 3}&{0 - 4} \\\ {0 - 1}&{0 - 2}&{0 - 3} \\\ {0 - x}&{0 - y}&{0 - z} \end{array}} \right| = 0
\therefore \left| {\begin{array}{*{20}{l}} { - 2}&{ - 3}&{ - 4} \\\ { - 1}&{ - 2}&{ - 3} \\\ { - x}&{ - y}&{ - z} \end{array}} \right| = 0
Now, we can remove the negative sign inside the determinant in the L.H.S of the above equation by multiplying each row by -1.
Thus, we have

2&3&4 \\\ 1&2&3 \\\ x&y;&z; \end{array}} \right| = 0$$ (I) Now, by solving the above determinant, we get, $$\therefore \left| {\begin{array}{*{20}{l}} 2&3&4 \\\ 1&2&3 \\\ x&y;&z; \end{array}} \right| = 2\left( {2z - 3y} \right) - 3\left( {z - 3x} \right) + 4\left( {y - 2x} \right)$$ $ = 4z - 6y - 3z + 9x + 4y - 8x$ $ = x - 2y + z$ (II) Now, by using equation (I) and (II), we get the equation of the plane as $x - 2y + z$ . **Hence, the equation of the plane is $x - 2y + z$.** **Note:** One of the properties of the determinant is as follows: Since, if we multiply any row or column of a determinant by a constant k, then the determinant gets multiplied by k. Determinant: The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted as $\det \left( A \right),\det A$ or $\left| A \right|$.