1
Question
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)
→r.(2^i+2^j−3^k)=7,→r.(2^i+5^j+3^k)=9 and through the point (2,1,3)
Open in App
Solution
The equations of the planes are
⇒ →r.(2^i+2^j−3^k)−7=0..........(1)
→r.(2^i+5^j+3^k)−9=0..........(2)
The equations of the planes through the intersection of the planes given in equations (1) and (2) is given by,
[→r.(2^i+2^j−3^k)−7]+λ[→r.(2^i+5^j+3^k)−9]=0, where λ∈R
→r.[(2^i+2^j−3^k)+λ(2^i+5^j+3^k)]=9λ+7
→r.[(2+2λ)^i+(2+5λ)^j+(3λ−3)^k]=9λ+7......(3)
The plane passes through the point (2,1,3). Therefore, its poisition vector is given by,
→r=2^i+^j+3^k
Substituting in equation (3), we obtain
(2^i+^j−3^k).[(2+2λ)^i+(2+5λ)^j+(3λ−3)^k]=9λ+7
⇒ 2(2+2λ)+(2+5λ)−3(3λ−3)=9λ+7
⇒ 9λ=8
⇒ 9λ=8⇒λ=89
Substituting λ=89 in equation (3), we obtain
→r.(349^i+589^j−39^k)=15
⇒ →r.(2^i+2^j−3^k)−7=0..........(1)
→r.(2^i+5^j+3^k)−9=0..........(2)
The equations of the planes through the intersection of the planes given in equations (1) and (2) is given by,
[→r.(2^i+2^j−3^k)−7]+λ[→r.(2^i+5^j+3^k)−9]=0, where λ∈R
→r.[(2^i+2^j−3^k)+λ(2^i+5^j+3^k)]=9λ+7
→r.[(2+2λ)^i+(2+5λ)^j+(3λ−3)^k]=9λ+7......(3)
The plane passes through the point (2,1,3). Therefore, its poisition vector is given by,
→r=2^i+^j+3^k
Substituting in equation (3), we obtain
(2^i+^j−3^k).[(2+2λ)^i+(2+5λ)^j+(3λ−3)^k]=9λ+7
⇒ 2(2+2λ)+(2+5λ)−3(3λ−3)=9λ+7
⇒ 9λ=8
⇒ 9λ=8⇒λ=89
Substituting λ=89 in equation (3), we obtain
→r.(349^i+589^j−39^k)=15
Suggest Corrections
0
and through the point (2, 1, 3)