(c) The planes $$ \Pi_{2} $$ and $$ \Pi_{3} $$ are with equations $$ \left.2 x+2 y+1\right) $$ respectively, where $$ p $$ is a constant. (i) Find the vector equation of the line of intersection between planes $$ \Pi_{1} $$ and $$ \Pi_{2} $$. (ii) Determine the value of $$ p $$ such that the planes $$ \Pi_{1}, \Pi_{2} $$ and $$ \Pi_{3} $$ do not have any common point.

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Mann Fuentes · Accepted Answer

(i) The vector equation of the line of intersection between planes $$ \Pi_{1} $$ and $$ \Pi_{2} $$ is: $$ \mathbf{r} = (0, 1, 0) + t(-2, 2, 0) $$ (ii) The value of $$ p $$ such that planes $$ \Pi_{1} $$, $$ \Pi_{2} $$, and $$ \Pi_{3} $$ do not have any common point is: $$ p \neq 1 $$

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