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Let

$$ a_1 x+b_1 y+c_1 z=d_1\\ a_2 x+b_2 y+c_2 z=d_2\\ a_3 x+b_3 y+c_3 z=d_3 $$

be three planes and it was given that if $d_1=d_2=d_3=1$ then the planes intersect exactly one point. Now my question is if we change the value of $d_1$, $d_2$, $d_3$ like if $d_1=2$, $d_2 =3$, $d_3=4$ do they still intersect at a unique point?

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1 Answer 1

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Yes, they will. One way to see this is because the matrix

$$ \left( \begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array} \right) $$

is invertible.

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