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I need some help with this problem can anyone give the solution and the formula needed / ways please ? I am stuck with this problem

(a) Find the vector equation for the plane containing the points $(x, y, z) = (1, −2, 3)$ and is parallel to the vectors $u = i + 2k$ and $v = 3i + j + k$. Then find the corresponding Cartesian equation for the plane.

(b) Find the vector equation for the line perpendicular to the plane in part (a) and passes through the point $(x, y, z) = (2, −1, −1)$.

(c) Find the Cartesian coordinates of the point of intersection between the line in part (b) and the plane in part (a).

(d) Find the minimum from the point $(x, y, z) = (2, −1, −1)$ and the plane in part (a)

Thanks,

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    $\begingroup$ Welcome to StackExchange! The site is not here to blindly answer questions which look like homework. Please show us some work you have already done towards the problem so that people know what your level of knowledge is and don't tell you things you already know $\endgroup$
    – lioness99a
    May 16 '17 at 9:06
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Some hints:

(a) See here. For a plane $\Pi$, given $p\in\Pi$ and two vectors tangent to $\Pi$, $T_1$ and $T_2$, one can parametrize the plane as: $$ \Pi(s,t) = p + sT_1 + tT_2 $$ so that one starts at $p$ and walks around the plane with distance given by $s,t$ in directions given by $T_i$. Just expand this to get the Cartesian version.

(b) Given two tangent vectors $T_1,T_2$ to a surface, a normal vector perpendicular to both is given by $$ n = T_1 \times T_2 $$ One can then construct a line from any direction vector $\vec{v}$ and intersecting point $q$ as $$ \ell(s) = q + s\vec{v} $$

(c) Equate $\ell$ and $\Pi$ using their equations and solve for the intersection.

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