Manipal Entrance Test 2016 Math Paper

The line of intersection of the planes r.(3

^

i

−

^

j

+

^

k

)=1 and r.(

^

i

−4

^

j

+2

^

k

)=2 is common to both the planes.
Therefore, it is perpendicular to normals to the two planes i.e. n1=3

^

i

−

^

j

+

^

k

and n2=

^

i

+4

^

j

−2

^

k

.
Hence, it is parallel to the vector n1×n2=−2

^

i

+7

^

j

+13

^

k

Thus, we have w find the equation of the plane passing through a=

^

i

+2

^

j

−

^

k

and normal to the vector n=n1×n2.
The equation of the required plane is
(r−a).n or r.n=a.n
or r.(−2

^

i

+7

^

j

+13

^

k

) =(

^

i

+2

^

j

+

^

k

). (−2

^

i

+7

^

j

+13

^

k

)
or r.(2

^

i

−7

^

j

−13

^

k

)=1