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Show that the line of intersection of the planes ` vec r.( hat i+2 hat j+3 hat k)=0` and `vec r.(3 hat i+2 hat j+ hat k)=0` is equally inclined to `ia n dkdot` Also find the angle it makes with `jdot`

Text Solution

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The line of intersection of the two planes will be perpendicular to the normals to the planes. Hence, it is parallel to the vector `(hati+2hatj+3hatk)xx(3hati+2hatj+hatk)=(-4hati+8hatj-4hatk)`. Now,
`" "(-4hati+8hatj-4hatk)*hati=-4 and (-4hati+8hatj-4hatk)*hatk=-4`
Hence, the line is equally inclined to `hati and hatk`. Also,
`" "((-4hati+8hatj-4hatk))/(sqrt(16+64+16))*hatj=(8)/(sqrt(96))=sqrt((2)/(3))`
If `theta` is the required angle, then `costheta=sqrt((2)/(3)) or theta = cos^(-1) sqrt((2)/(3))`.
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