Let p1=[1 0 2] , p2=[-1 2 1] , n1=[1 1 -2] , and n2=[-2 1 3] ; let H1 be the hyperplane (plane) in R^3 passing through the point p1 and having normal vector n1 ; and let H2 be the hyperplane passing through the point p2 and having normal vector n2 . Give an explicit description of H1∩H2 by a formula

Question

Let [latex] p_{1} = \left [ \begin{matrix} 1 \ 0 \ 2 \end{matrix} ight ] , p_{2} = \left [ \begin{matrix} -1 \ 2 \ 1 \end{matrix} ight ] , n_{1} = \left [ \begin{matrix} 1 \ 1 \ -2 \end{matrix} ight ] , ext{and} n_{2} = \left [ \begin{matrix} -2 \ 1 \ 3 \end{matrix} ight ] [/latex] ; let [latex] H_{1} [/latex] be the hyperplane (plane) in [latex] \mathbb{R} ^{3} [/latex] passing through the point [latex] p_{1} [/latex] and having normal vector [latex] n_{1} [/latex]; and let [latex] H_{2} [/latex] be the hyperplane passing through the point [latex] p_{2} [/latex] and having normal vector [latex] n_{2} [/latex]. Give an explicit description of [latex] H_{1} \cap H_{2} [/latex] by a formula that shows how to generate all points in [latex] H_{1} \cap H_{2} [/latex].

Accepted Answer

First, compute [latex] n_{1} \cdot p_{1} = -3 [/latex] and [latex] n_{2} \cdot p_{2} = 7 [/latex]. The hyperplane [latex] H_{1} [/latex] is the solution set of the equation [latex] x_{1} + x_{2} - 2x_{3} = -3 [/latex], and [latex] H_{2} [/latex] is the solution set of the equation [latex] -2x_{1} + x_{2} + 3x_{3} = 7 [/latex]. Then

[latex] H_{1} \cap H_{2} = \{x : x_{1} + x_{2} - 2x_{3} = -3 ext{and} -2x_{1} + x_{2} + 3x_{3} = 7\} [/latex]

This is an implicit description of [latex] H_{1} \cap H_{2} [/latex]. To find an explicit description, solve the system of equations by row reduction:

[latex] \left [ \begin{matrix} 1 & 1 & -2 & -3 \ -2 & 1 & 3 & 7 \end{matrix} ight ] \sim \left [ \begin{matrix} 1 & 0 & -\frac{5}{3} & -\frac{10}{3} \ 0 & 1 & -\frac{1}{3} & \frac{1}{3} \end{matrix} ight ] [/latex]

Thus [latex] x_{1} = -\frac{10}{3} + \frac{5}{3}x_{3} , x_{2} = \frac{1}{3} + \frac{1}{3}x_{3} , x_{3} = x_{3} [/latex]. Let [latex] p = \left [ \begin{matrix} -\frac{10}{3} \ \frac{1}{3} \ 0 \end{matrix} ight ] ext{and} v = \left [ \begin{matrix} \frac{5}{3} \ \frac{1}{3} \ 1 \end{matrix} ight ] [/latex] . The general solution can be written as [latex] x = p + x_{3}v [/latex]. Thus [latex] H_{1} \cap H_{2} [/latex] is the line through p in the direction of v. Note that v is orthogonal to both [latex] n_{1} [/latex] and [latex] n_{2} [/latex].

Question 8.4.PP.1: Let p1=[1 0 2] , p2=[-1 2 1] , n1=[1 1 -2] , and n2=[-2 1 3]...

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