Find the equation of the plane passing through the point $(2, - 1, 1)$ and through the line of intersection of the planes $\to r \cdot (2^i - 3^j +^k) = 3$ and $\to r \cdot (^i + 5^j -^k) = 4.4$ .

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Solution

We know that equation of plane passing through the line of intersection of the planes is
$\to r . (\to a + λ \to b) = c_{1} + λ c_{2}$
where $\to a$ and $\to b$ are the normal vectors of the planes and $c_{1}$ and $c_{2}$ are the intercept terms of the respective planes.

So, here, equation of the required plane is,
$\to r . ((2 + λ)^i + (- 3 + 5 λ)^j + (1 - λ)^k) = 3 + 4.4 λ$

It is given that $(2, - 1, 1)$ passes through the plane.

$⟹ (2^i -^j +^k) . ((2 + λ)^i + (- 3 + 5 λ)^j + (1 - λ)^k) = 3 + 4.4 λ$
$⟹ 2 (2 + λ) - (- 3 + 5 λ) + (1 - λ) = 3 + 4.4 λ$
$4 + 2 λ + 3 - 5 λ + 1 - λ = 3 + 4.4 λ$
$8.4 λ = 5$

Substituting $λ = 5 / 8.4$ we get the equation as,
$\to r . ((21.8)^i + (- 0.2)^j + (3.4)^k) = 47.2$

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Find the equation of the plane passing through the point (2,−1,1) and through the line of intersection of the planes →r⋅(2^i−3^j+^k)=3 and →r⋅(^i+5^j−^k)=4.4.

Find the equation of the plane passing through the point $(2, - 1, 1)$ and through the line of intersection of the planes $\to r \cdot (2^i - 3^j +^k) = 3$ and $\to r \cdot (^i + 5^j -^k) = 4.4$ .