Find the vector and Cartesian equations of the plane passing through the points $(2, 2, - 1), (3, 4, 2) and (7, 0, 6) .$ Also find the vector equation of a plane passing through $(4, 3, 1)$ and parallel to the plane obtained above.

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Solution

The equation of the plane which passes through given points is given by
$∣ ∣ ∣ ∣ \begin{matrix} x - 2 & y - 2 & z + 1 3 - 2 & 4 - 2 & 2 + 1 7 - 2 & 0 - 2 & 6 + 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} x - 2 & y - 2 & z + 1 1 & 2 & 3 5 & - 2 & 7 \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow (x - 2) (14 + 6) - (y - 2) (7 - 15) + (z + 1) (- 2 - 10) = 0 \Rightarrow 5 x + 2 y - 3 z = 17$

The vector equation of the given plane is $\to r . (5^i + 2^j - 3^k) = 17$
The equation of the plane which is parallel to the above plane is given by
$\to r . (5^i + 2^j - 3^k) = λ$
$Since, it passes through (4, 3, 1) (4^i + 3^j +^k) \cdot (5^i + 2^j - 3^k) = λ \Rightarrow λ = 23$
Therefore, the equation of the required plane is $\to r . (5^i + 2^j - 3^k) = 23$

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