Question

Find the equation of the right bisector plane of the segment joining (2, 3, 4) and (6, 7, 8)

Answer 1

Hint: A right bisector is a line that cuts another line at midpoint at 90 degrees. It is more often called a perpendicular bisector. To solve this we need to know the formula for Cartesian equation of a line passes through two points \[({x_1},{y_1},{z_1}) \] and \[({x_{2,}}{y_2},{z_2}) \] . Also remember the basic definition of direction cosines of a line.

Complete step-by-step answer:
If we describe the given problem in diagram, we get

We know the Cartesian equation of a line passes through two points \[({x_1},{y_1},{z_1}) \] and \[({x_{2,}}{y_2},{z_2}) \] is:
 \[ \begin{gathered}
   \dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}} \ \
     \ \
 \end{gathered} \] .
Here, \[({x_1},{y_1},{z_1}) \] = (2, 3, 4) and \[({x_{2,}}{y_2},{z_2}) \] = (6, 7, 8)
Substituting this in above,
 \[ \Rightarrow \dfrac{{x - 2}}{{6 - 2}} = \dfrac{{y - 3}}{{7 - 3}} = \dfrac{{z - 4}}{{8 - 4}} \]
 \[ \Rightarrow \dfrac{{x - 2}}{4} = \dfrac{{y - 3}}{4} = \dfrac{{z - 4}}{4} \] .
Which is equivalent to
 \[ \begin{gathered}
    \Rightarrow \dfrac{{x - 2}}{1} = \dfrac{{y - 3}}{1} = \dfrac{{z - 4}}{1} \ \
     \ \
 \end{gathered} \]
Therefore direction cosines are (1, 1, 1).
The bisector point is given C is given by, \[ \Rightarrow \left( { \dfrac{{{x_1} + {x_2}}}{2}, \dfrac{{{y_1} + {y_2}}}{2}, \dfrac{{{z_1} + {z_2}}}{2}} \right) \]
Substituting \[({x_1},{y_1},{z_1}) \] = (2, 3, 4) and \[({x_{2,}}{y_2},{z_2}) \] = (6, 7, 8)
 \[ \Rightarrow \left( { \dfrac{{2 + 6}}{2}, \dfrac{{3 + 7}}{2}, \dfrac{{4 + 8}}{2}} \right) \]
 \[ \Rightarrow (4,5,6) \]
We get the point C (4, 5, 6).
Hence the equation of the right bisector in a plane is \[l(x - {c_1}) + m(y - {c_2}) + n(z - {c_3}) = 0 \] .
Where, \[(l,m,n) \] are direction cosines and \[({c_1},{c_2},{c_3}) = (4,5,6) \] .
Substituting above we get,
 \[1(x - 4) + 1(y - 5) + 1(z - 6) = 0 \]
Keeping the variable on one side and constant on the other side,
 \[ \Rightarrow x + y + z = 4 + 5 + 6 \]
 \[ \Rightarrow x + y + z = 15 \]
Hence, the equation of the right bisector plane of the segment joining (2, 3, 4) and (6, 7, 8) is \[x + y + z = 15 \]
So, the correct answer is “ \[x + y + z = 15 \] ”.

Note: In this type of question we need to remember the Cartesian equation of a line passes through two points \[({x_1},{y_1},{z_1}) \] and \[({x_{2,}}{y_2},{z_2}) \] . Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes. Also remember the equation of the bisector in a plane. Which is the same for any problem, so that you can solve for different points.