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Equation of a Plane : Vector Form

The vector eq...

Question

The vector equation of the plane passing through the intersection of the planes
$\to r \cdot (2^i + 3^j - 3^k) = 7,$ $\to r \cdot (2^i + 5^j + 3^k) = 9$ and through the point $(2, 1, 3)$ is $\to r \cdot (^i + 2^j) = p$ then $p$ is

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Solution

Given: $p_{1} : 2 x + 3 y - 3 z - 7 = 0$ and
$p_{2} : 2 x + 5 y + 3 z - 9 = 0$
Any plane through the line of intersection is
$p_{1} + λ p_{2} = 0$
Intersection of given plane is
$(2 x + 3 y - 3 z - 7) + λ (2 x + 5 y + 3 z - 9) = 0$

Given, It is passing through $(2, 1, 3)$

$∴ (4 + 3 - 9 - 7) + λ (4 + 5 + 9 - 9) = 0 - 9 + 9 λ = 0 ∴ λ = 1$

Hence, equation of plane is $4 x + 8 y - 16 = 0 = x + 2 y - 4 = 0$

In vector form $\to r \cdot (^i + 2^j) = 4$
$∴ p = 4$

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