First slide

Planes in 3D

Question

$Consider the planes P_{1} : 2 x + y + z + 4 = 0, P_{2} : y - z + 4 = 0 and P_{3} : 3 x + 2 y + z + 8 = 0 . Let L_{1}$
$L_{2}, L_{3} be the lines of intersection of the planes P_{2} and P_{3}, P_{3} and P_{1}, and P_{1} and P_{2} respectively . then$

Moderate

A

$Atleast two of the lines L_{1}, L_{2} and L_{3} are non-parallel$
B

$Atleast two of the lines L_{1}, L_{2} and L_{3} are parallel$
C

$The three planes intersect in a line$
D

$The three planes form a triangular prism$

Solution

$Observe that the lines L_{1}, L_{2} & L_{3} are parallel$ $to the vector \hat{i} - \hat{j} - \hat{k}$
$Also, Δ = 0 = Δ_{1} & b_{1} c_{2} \neq b_{1} c_{1} Hence the three planes intersect in a line.$
$\begin{array}{l} P_{1} = 2 x + y + z + 4 = 0 \\ P_{2} = 0 x + y - z + 4 = 0 \\ P_{3} = 3 x + 2 y + z + 8 = 0 \\ P_{2} and P_{3} gives line L_{1} \end{array}$
$Vector parallel to line L_{1} |\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & - 1 \\ 3 & 2 & 1 \end{array}|$
$\begin{aligned} = 3 \hat{i} - 3 \hat{j} - 3 \hat{k} \\ = 3 [\hat{i} - \hat{j} - \hat{k}] \end{aligned}$
Similarly
$Vector parallel to L_{2}, P_{3} and P_{1} = |\begin{array}{lll} \hat{i} \hat{j} \hat{k} \\ 2 1 1 \\ 3 2 1 \end{array}|$ $\begin{aligned} = - 2 \hat{i} + 2 \hat{j} + \hat{k} \\ = - 2 (\hat{i} - \hat{j} - \hat{k}) \end{aligned}$

$= - \hat{i} + \hat{j} + \hat{k} = - [\hat{i} - \hat{j} - \hat{k}]$
Similarly
$Vector parallel to L_{3}, P_{1} and P_{2} = |\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & 1 \\ 0 & 1 & - 1 \end{array}|$
$= (- 2) \hat{i} - (- 2) \hat{j} + 2 \hat{k}$
We can see all the lines are parallel to vector
$\begin{array}{l} (\hat{i} - \hat{j} - \hat{k}) \\ Also 2 x + y + z = - 4 \\ 0 x + y - z = - 4 \\ 3 x + 2 y + z = - 8 \\ Δ = |\begin{array}{ccc} 2 & 1 & 1 \\ 0 & 1 & - 1 \\ 3 & 2 & 1 \end{array}| \\ \Rightarrow 2 (1 + 2) - 1 (0 + 3) + 1 (0 - 3) \\ Δ_{2} = |\begin{array}{ccc} 2 & - 4 & 1 \\ 0 & - 4 & - 1 \\ 3 & - 8 & 1 \end{array}| = 0 \\ = - 4 |\begin{array}{ccc} 2 & 1 & 1 \\ 0 & 1 & - 1 \\ 3 & 2 & 1 \end{array}| = 0 \\ Δ_{3} = 0 \end{array}$
So all planes intersection in line L.

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