Find the vector equation of the plane passing through the intersection of the planes 2- i -2- j -3- k -7- 2- i -5- j -3- k -9 and through the point 2-1-3

[ π _ 1 =r⃗( 2î +2ĵ 3k̂ #160; ) =7; ...

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Standard XII

Mathematics

Family of Planes Passing through the Intersection of Two Planes

Find the vect...

Question

Find the vector equation of the plane passing through the intersection of the planes $r \to (2 \land i + 2 \land j - 3 \land k) = 7, r \to (2 \land i + 5 \land j + 3 \land k) = 9$ and through the point (2,1,3)

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Solution

$\begin{matrix} π_{1} = \to r (2^i + 2^j - 3^k) = 7 \Rightarrow 2 x + 2 y - 3 z = 7 π_{2} = \to r (2^i + 5^j + 3^k) = 9 \Rightarrow 2 x + 5 y + 3 z = 9 p l a n e t h r o u g h l i n e o f i n t e r s e c t o i o n o f π_{1} a n d π_{2} i s : (2 x + 2 y - 3 z - 7) + λ (2 x + 5 y + 3 z - 9) = 0 (2 + 2 λ) x + y (2 + 5 λ) + z (- 3 + 3 λ) - 7 - 9 λ = 0 2 (2 + 2 λ) + 2 + 5 λ + 3 (- 3 + 3 λ) - 7 - 9 λ = 0 - 10 + 9 λ = 0 λ = \frac{10}{9} (2 + \frac{20}{9}) x + y (2 + \frac{50}{9}) + z (- 3 + \frac{30}{9}) - 7 - 9 \times \frac{10}{9} = 0 \frac{38 x}{9} + \frac{68 y}{9} + \frac{3 z}{9} = 17 38 x + 68 y + 3 z = 153 \end{matrix}$

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Find the vector equation of the plane passing through the intersection of the planes $r . (2^i + 2^j - 3^k) = 7, r . (2^i + 5^j + 3^k) = 9$ and through the point (2,1,3).

Q. Find the equation of the plane passing through the intersection of the planes

\vec{r} \cdot (2 \hat{i} + \hat{j} + 3 \hat{k}) = 7, \vec{r} \cdot (2 \hat{i} + 5 \hat{j} + 3 \hat{k}) = 9

and the point (2, 1, 3).

Q. 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)

\to r \cdot (^i + 2^j) = p

then

p

Q. Find the vector equation of the plane passing through the intersection of the planes

\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 and \vec{r} \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = - 5

and the point (1, 1, 1).

Q. Find the cartesion equation of the plane passing through the point

(- 2, 1, 3)

and perpendicular to the vector

3 i + j + 5 k .

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Find the vector equation of the plane passing through the intersection of the planes r→(2∧i+2∧j−3∧k)=7,r→(2∧i+5∧j+3∧k)=9 and through the point (2,1,3)

Find the vector equation of the plane passing through the intersection of the planes $r \to (2 \land i + 2 \land j - 3 \land k) = 7, r \to (2 \land i + 5 \land j + 3 \land k) = 9$ and through the point (2,1,3)