Let ∏1, ∏2, and ∏3 be planes having the following equations: Equation of ∏1: 6x + 4y + 8z = 2 Equation of ∏2: x + 5y + 3z = 4 Equation of ∏3: 3x + 2y + 4z = 9 (a) Determine if any two of these three planes are parallel. (b) Find the equation of the plane parallel to ∏1, passing through

Question

Let [latex]\Pi_{1}, \Pi_{2}, ext { and } \Pi_{3}[/latex] be planes having the following equations:

[latex] ext { Equation of } \Pi_{1}: \quad 6 x+4 y+8 z=2[/latex]
[latex] ext { Equation of } \Pi_{2}: \quad x+5 y+3 z=4[/latex]
[latex] ext { Equation of } \Pi_{3}: \quad 3 x+2 y+4 z=9[/latex]

(a) Determine if any two of these three planes are parallel.

(b) Find the equation of the plane parallel to [latex]\Pi_{1}[/latex], passing through [latex]P_{0}=(1,2.1)[/latex].

Accepted Answer

(a) Let [latex]n _{1}, n _{2}, ext { and } n _{3}[/latex] denote the normals for [latex]\Pi_{1}, \Pi_{2}, ext { and } \Pi_{3}[/latex], respectively. Then

[latex]n _{1}=\left[\begin{array}{l}6 \4 \8\end{array} ight], \quad n _{2}=\left[\begin{array}{l}1 \5 \3\end{array} ight], \quad n _{3}=\left[\begin{array}{l}3 \2 \4\end{array} ight][/latex].

Clearly, [latex]n _{1}=2 n _{3}, ext { and so } n _{1} ext { and } n _{3}[/latex] are parallel vectors. Therefore, [latex]\Pi_{1}[/latex] and [latex]\Pi_{3}[/latex] are parallel planes.

(b) Any plane parallel to [latex]\Pi_{1}[/latex] has a normal equal to [latex]n _{1}[/latex]. Thus, using Eq. (1) [latex]A\left(x-x_{0} ight)+B\left(y-y_{0} ight)+C\left(z-z_{0} ight)=0[/latex], we find the equation of the plane parallel to [latex]\Pi_{1}[/latex] and containing [latex]P_{0}=(1,2,1)[/latex] has equation
6x + 4y + 8z = 22
or, after canceling a common factor of 2,

3x + 2y + 4z = 11.

Question 2.4.8: Let ∏1, ∏2, and ∏3 be planes having the following equations:...

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