First slide

Planes in 3D

Question

If $P_{1} : \vec{r} \cdot {\vec{n}}_{1} - d_{1} = 0, P_{2} : \vec{r} \cdot \vec{n_{2}} - d_{2} = 0$ and $P_{3} : \vec{r} \cdot \vec{n_{3}} - d_{3} = 0$ are three planes $\vec{n_{1}}, \vec{n_{2}} and \vec{n_{3}}$ are three non-coplanar vectors, then three lines P₁ =0, P₂=0; P₂= 0, P₃ = 0 and P₃ : 0, P₁ : 0 are

Moderate

A

parallel lines
B

coplanar lines
C

coincident lines
D

concurrent lines

Solution

P₁ =0, P₂=0; P₂= 0, P₃ = 0 and P₃ : 0, P₁ : 0 are lines of intersection of the three planes P ₁ P₂ and P₃
As $\vec{n_{1}}, \vec{n_{2}} and \vec{n_{3}}$ are non-coplanar, planes P₁, P₂and P₃ will intersect at unique point.
So the given lines will pass through a fixed point.

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