Chapter 11, Problem 14CLT

To determine

To Calculate:

Parametric form of equation.

Answer to Problem 14CLT

The parametric function of equation of $\left(-1,2,0\right)$ is,

  $\begin{array}{l}x=-1+2t\\ y=2-4t\\ z=t\end{array}$

Explanation of Solution

Given Information:

Points: $\left(-1,2,0\right)$

Equation: $2x-4y+z=8$

Calculation:

Consider a point and an equation of the plane:

  $\left(-1,2,0\right)$ and $2x-4y+z=8$

To determine the required parametric equation of the line, proceed as follows.

Find the normal vector of the plane using standard result.

Normal vector to plane whose equation is $\langle 2,-4,1\rangle$

As the required line perpendicular to the plane, the required line will be parallel to the normal vector of the plane.

Now use the standard result of vector equation of the line passing through a given point and parallel to vector.

Point $\left(a_1,a_2,a_3\right)$

Parallel vector: $\langle b_1,b_2,b_3\rangle$

Vector equation: $r=\langle a_1,a_2,a_3\rangle +i\langle b_1,b_2,b_3\rangle$

Hence, the vector equation of the required line is

  $r=\langle -1,2,0\rangle +t\langle 2,-4,1\rangle$

Now use the equality of vectors and express the vector equation into its parametric equation.

  $\begin{array}{l}r=\langle -1,2,0\rangle +t\langle 2,-4,1\rangle \\ \left(x,y,z\right)=\langle -1,2,0\rangle +t\langle 2,-4,1\rangle \\ =\langle -1+2t,2-4t,t\rangle \end{array}$

Therefore,

  $\begin{array}{l}x=-1+2t\\ y=2-4t\\ z=t\end{array}$

Hence, the parametric equation of the required line is

  $\begin{array}{l}x=-1+2t\\ y=2-4t\\ z=t\end{array}$