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Find an equation of the hyperplane $H$ in $\mathbf{R}^{4}$ that passes through the point $P(1,3,-4,2)$ and is normal to the vector $u=[4,-2,5,6]$.
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The coefficients of the unknowns of an equation of $H$ are the components of the normal vector $u$; hence, the equation of $H$ must be of the form
$$4 x_{1}-2 x_{2}+5 x_{3}+6 x_{4}=k$$
Substituting $P$ into this equation, we obtain
$4(1)-2(3)+5(-4)+6(2)=k \quad$ or $\quad 4-6-20+12=k \quad$ or $\quad k=-10$
Thus, $4 x_{1}-2 x_{2}+5 x_{3}+6 x_{4}=-10$ is the equation of $H$.
by Bronze Status (5,550 points)

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