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)


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    – lioness99a
    May 16 '17 at 9:06

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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