Question

Find an equation of the plane contains the line \({\rm{x = 1 + t, y = 2 - t, z = 4 - 3 t}}\)and parallel to the plane \({\rm{5 x + 2 y + z = 1}}.\)

Short answer

The equation of the plane that contains the line \({\rm{x = 1 + t, y = 2 - t, z = 4 - 3 t}}\) and parallel to the plane \(5x + 2y + z = 1\)is\(5x + 2y + z = 13\).

Step-by-step solution

Expression to find an equation of the plane.

Write an expression to find an equation of the plane through the point\({P_0}\left( {{x_0},{y_0},{z_0}} \right)\)and with the normal vector\({\rm{n}} = \langle a,b,c\rangle \)as follows.

\(a\left( {x - {x_0}} \right) + b\left( {y - {y_0}} \right) + c\left( {z - {z_0}} \right) = 0.\) …… (1)

The normal vector of the plane\(5x + 2y + z = 1\)is\(\langle 5,2,1\rangle {\rm{. }}\)

As the two planes are in parallel, the normal vectors of both the planes are equal.

Therefore, the normal vector of the plane that contains the line\({\rm{x = 1 + t, y = 2 - t, z = 4 - 3 t}}\)is also\(\langle 5,2,1\rangle {\rm{. }}\)

\({\rm{n}} = \langle 5,2,1\rangle \)

The parametric equations of the line are written as follows.

\({\rm{x = 1 + t, y = 2 - t, z = 4 - 3 t}}\) …… (2)

Write the expression for the parametric equations for the line of the point\(\left( {{x_0},{y_0},{z_0}} \right)\)and parallel to the direction vector\(\langle a,b,c\rangle .\)

\({\rm{x = x0 + a t, y = y0 + b t, z = z0 + c t}}{\rm{.}}\) …… (3)

Use the expression for calculation.

Compare equation (3) with (2) and write the point at which the line passes.

\(\left( {{x_0},{y_0},{z_0}} \right) = (1,2,4)\)

Therefore, the plane passes through the point\((1,2,4)\) and has the normal vector of\(\langle 5,2,1\rangle {\rm{. }}\)

Substitute\(1{\rm{ for }}{x_0},2{\rm{ for }}{y_0},4{\rm{ for }}{z_0},5{\rm{ for }}a,2{\rm{ for }}b{\rm{,}}\) and\(1{\rm{ for c}}\) in equation (1),

\(\begin{array}{l}(5)(x - 1) + (2)(y - 2) + (1)(z - 4) = 0\\5x - 5 + 2y - 4 + z - 4 = 0\\5x + 2y + z = 13\end{array}\)

Thus, the equation of the plane that contains the line \({\rm{x = 1 + t, y = 2 - t, z = 4 - 3 t}}\) and parallel to the plane \(5x + 2y + z = 1\)is\(5x + 2y + z = 13\).