JEE Main Math Class 12 Three Dimensional Geometry Part 1 Questions

Normal to 3x+4y+z=1 is 3

^

i

+4

^

j

+

^

k

.
Normal to 5x+8y+2z=−14 is 5

^

i

+8

^

j

+2

^

k

The line of intersection of the planes is perpendicular to both normals, so, direction ratios of the intersection line are directly proportional to the cross product of the normal vectors.
Therefore the direction ratios of the line is −

^

j

+4

^

k

Hence the angle between the plane x+y+z+5=0 and the intersection line is sin−1(

−1+4

√17√3

)=sin−1(√

3

17

)