The question
Find the vector equation of the plane passing through the intersection of the planes r ⃗ ⋅ ( 2 i ^ + 2 j ^ − 3 k ^ ) = 7 \vec r\cdot(2\hat i + 2\hat j - 3\hat k) = 7 r ⋅ ( 2 i ^ + 2 j ^ − 3 k ^ ) = 7 r ⃗ ⋅ ( 2 i ^ + 5 j ^ + 3 k ^ ) = 9 \vec r\cdot(2\hat i + 5\hat j + 3\hat k) = 9 r ⋅ ( 2 i ^ + 5 j ^ + 3 k ^ ) = 9 ( 2 , 1 , 3 ) (2, 1, 3) ( 2 , 1 , 3 )
This topic is no longer in the CBSE syllabus (2025-26).
Exam answer Show all explanations
The equation of a plane through the intersection of the given planes is
r ⃗ ⋅ ( 2 i ^ + 2 j ^ − 3 k ^ ) − 7 + λ [ r ⃗ ⋅ ( 2 i ^ + 5 j ^ + 3 k ^ ) − 9 ] = 0 \begin{aligned} &\vec r\cdot(2\hat i + 2\hat j - 3\hat k) - 7 \\ &\quad + \lambda[\vec r\cdot(2\hat i + 5\hat j + 3\hat k) - 9] = 0 \end{aligned} r ⋅ ( 2 i ^ + 2 j ^ − 3 k ^ ) − 7 + λ [ r ⋅ ( 2 i ^ + 5 j ^ + 3 k ^ ) − 9 ] = 0 Why?
For planes r ⃗ ⋅ n ⃗ 1 = d 1 \vec r\cdot\vec n_1 = d_1 r ⋅ n 1 = d 1 r ⃗ ⋅ n ⃗ 2 = d 2 \vec r\cdot\vec n_2 = d_2 r ⋅ n 2 = d 2
( r ⃗ ⋅ n ⃗ 1 − d 1 ) + λ ( r ⃗ ⋅ n ⃗ 2 − d 2 ) = 0 \begin{aligned} &(\vec r\cdot\vec n_1 - d_1) \\ &\quad + \lambda(\vec r\cdot\vec n_2 - d_2) = 0 \end{aligned} ( r ⋅ n 1 − d 1 ) + λ ( r ⋅ n 2 − d 2 ) = 0 A point on both planes makes both brackets zero, so it lies on this plane for every λ \lambda λ λ \lambda λ
or
r ⃗ ⋅ [ ( 2 + 2 λ ) i ^ + ( 2 + 5 λ ) j ^ + ( − 3 + 3 λ ) k ^ ] = 7 + 9 λ … ( i ) \begin{aligned} &\vec r\cdot[(2 + 2\lambda)\hat i + (2 + 5\lambda)\hat j \\ &\quad + (-3 + 3\lambda)\hat k] \\ &= 7 + 9\lambda \quad\ldots(\text{i}) \end{aligned} r ⋅ [( 2 + 2 λ ) i ^ + ( 2 + 5 λ ) j ^ + ( − 3 + 3 λ ) k ^ ] = 7 + 9 λ … ( i ) Why?
The dot product distributes over addition:
r ⃗ ⋅ n ⃗ 1 + λ r ⃗ ⋅ n ⃗ 2 = r ⃗ ⋅ ( n ⃗ 1 + λ n ⃗ 2 ) \begin{aligned} &\vec r\cdot\vec n_1 + \lambda\,\vec r\cdot\vec n_2 \\ &= \vec r\cdot(\vec n_1 + \lambda\vec n_2) \end{aligned} r ⋅ n 1 + λ r ⋅ n 2 = r ⋅ ( n 1 + λ n 2 ) Add matching components, for example 2 + 2 λ 2 + 2\lambda 2 + 2 λ i ^ \hat i i ^
( 2 i ^ + 2 j ^ − 3 k ^ ) + λ ( 2 i ^ + 5 j ^ + 3 k ^ ) = ( 2 + 2 λ ) i ^ + ( 2 + 5 λ ) j ^ + ( − 3 + 3 λ ) k ^ \begin{aligned} &(2\hat i + 2\hat j - 3\hat k) \\ &\quad + \lambda(2\hat i + 5\hat j + 3\hat k) \\ &= (2 + 2\lambda)\hat i + (2 + 5\lambda)\hat j \\ &\quad + (-3 + 3\lambda)\hat k \end{aligned} ( 2 i ^ + 2 j ^ − 3 k ^ ) + λ ( 2 i ^ + 5 j ^ + 3 k ^ ) = ( 2 + 2 λ ) i ^ + ( 2 + 5 λ ) j ^ + ( − 3 + 3 λ ) k ^ The constants − 7 − 9 λ -7 - 9\lambda − 7 − 9 λ 7 + 9 λ 7 + 9\lambda 7 + 9 λ
The point ( 2 , 1 , 3 ) (2, 1, 3) ( 2 , 1 , 3 )
2 ( 2 + 2 λ ) + 1 ( 2 + 5 λ ) + 3 ( − 3 + 3 λ ) = 7 + 9 λ \begin{aligned} &2(2 + 2\lambda) + 1(2 + 5\lambda) \\ &\quad + 3(-3 + 3\lambda) = 7 + 9\lambda \end{aligned} 2 ( 2 + 2 λ ) + 1 ( 2 + 5 λ ) + 3 ( − 3 + 3 λ ) = 7 + 9 λ Why?
Put r ⃗ = 2 i ^ + j ^ + 3 k ^ \vec r = 2\hat i + \hat j + 3\hat k r = 2 i ^ + j ^ + 3 k ^
⇒ 18 λ − 3 = 7 + 9 λ \Rightarrow 18\lambda - 3 = 7 + 9\lambda ⇒ 18 λ − 3 = 7 + 9 λ ⇒ 9 λ = 10 ⇒ λ = 10 9 \Rightarrow 9\lambda = 10 \Rightarrow \lambda = \frac{10}{9} ⇒ 9 λ = 10 ⇒ λ = 9 10 Why?
Expanding the left side:
2 ( 2 + 2 λ ) = 4 + 4 λ 1 ( 2 + 5 λ ) = 2 + 5 λ 3 ( − 3 + 3 λ ) = − 9 + 9 λ \begin{aligned} 2(2 + 2\lambda) &= 4 + 4\lambda \\ 1(2 + 5\lambda) &= 2 + 5\lambda \\ 3(-3 + 3\lambda) &= -9 + 9\lambda \end{aligned} 2 ( 2 + 2 λ ) 1 ( 2 + 5 λ ) 3 ( − 3 + 3 λ ) = 4 + 4 λ = 2 + 5 λ = − 9 + 9 λ 4 + 4 λ + 2 + 5 λ − 9 + 9 λ = 18 λ − 3 \begin{aligned} &4 + 4\lambda + 2 + 5\lambda - 9 + 9\lambda \\ &= 18\lambda - 3 \end{aligned} 4 + 4 λ + 2 + 5 λ − 9 + 9 λ = 18 λ − 3 Collect like terms: 4 λ + 5 λ + 9 λ = 18 λ 4\lambda + 5\lambda + 9\lambda = 18\lambda 4 λ + 5 λ + 9 λ = 18 λ 4 + 2 − 9 = − 3 4 + 2 - 9 = -3 4 + 2 − 9 = − 3
Then 18 λ − 9 λ = 7 + 3 18\lambda - 9\lambda = 7 + 3 18 λ − 9 λ = 7 + 3 9 λ = 10 9\lambda = 10 9 λ = 10
18 λ − 3 = 7 + 9 λ 18 λ − 9 λ = 7 + 3 9 λ = 10 λ = 10 9 \begin{aligned} 18\lambda - 3 &= 7 + 9\lambda \\ 18\lambda - 9\lambda &= 7 + 3 \\ 9\lambda &= 10 \\ \lambda &= \frac{10}{9} \end{aligned} 18 λ − 3 18 λ − 9 λ 9 λ λ = 7 + 9 λ = 7 + 3 = 10 = 9 10 Putting λ = 10 9 \lambda = \frac{10}{9} λ = 9 10
r ⃗ ⋅ ( 38 9 i ^ + 68 9 j ^ + 3 9 k ^ ) = 7 + 10 = 17 \begin{aligned} &\vec r\cdot\left(\frac{38}{9}\hat i + \frac{68}{9}\hat j + \frac39\hat k\right) \\ &= 7 + 10 = 17 \end{aligned} r ⋅ ( 9 38 i ^ + 9 68 j ^ + 9 3 k ^ ) = 7 + 10 = 17 Why?
Each component with λ = 10 9 \lambda = \frac{10}{9} λ = 9 10
2 + 2 ( 10 9 ) = 2 + 20 9 = 18 9 + 20 9 = 38 9 2 + 5 ( 10 9 ) = 2 + 50 9 = 18 9 + 50 9 = 68 9 − 3 + 3 ( 10 9 ) = − 3 + 30 9 = − 27 9 + 30 9 = 3 9 \begin{aligned} 2 + 2\left(\frac{10}{9}\right) &= 2 + \frac{20}{9} \\ &= \frac{18}{9} + \frac{20}{9} = \frac{38}{9} \\[4pt] 2 + 5\left(\frac{10}{9}\right) &= 2 + \frac{50}{9} \\ &= \frac{18}{9} + \frac{50}{9} = \frac{68}{9} \\[4pt] -3 + 3\left(\frac{10}{9}\right) &= -3 + \frac{30}{9} \\ &= -\frac{27}{9} + \frac{30}{9} = \frac39 \end{aligned} 2 + 2 ( 9 10 ) 2 + 5 ( 9 10 ) − 3 + 3 ( 9 10 ) = 2 + 9 20 = 9 18 + 9 20 = 9 38 = 2 + 9 50 = 9 18 + 9 50 = 9 68 = − 3 + 9 30 = − 9 27 + 9 30 = 9 3 and 9 λ = 10 9\lambda = 10 9 λ = 10 17 × 9 = 153 17 \times 9 = 153 17 × 9 = 153
9 × 38 9 = 38 9 × 68 9 = 68 9 × 3 9 = 3 9 × 17 = 153 \begin{aligned} 9 \times \frac{38}{9} &= 38 \\ 9 \times \frac{68}{9} &= 68 \\ 9 \times \frac39 &= 3 \\ 9 \times 17 &= 153 \end{aligned} 9 × 9 38 9 × 9 68 9 × 9 3 9 × 17 = 38 = 68 = 3 = 153 ∴ \therefore ∴ r ⃗ ⋅ ( 38 i ^ + 68 j ^ + 3 k ^ ) = 153 \vec r\cdot(38\hat i + 68\hat j + 3\hat k) = 153 r ⋅ ( 38 i ^ + 68 j ^ + 3 k ^ ) = 153