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Find the vector equation of the line p...

Find the vector equation of the line passing through `(1,2,3)` and parallel to the planes ` vec r.( hat i- hat j+2 hat k)=5a n d vec r.(3 hat i+ hat j+ hat k)=6.`

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Line is parallel to the planes `vecr*(hati-hatj+2hatk)=5 and vecr*(3hati+hatj+hatk)=6`, i.e., line is perpendicular to the vectors `hati-hatj+2hatk and 3hati+hatj+hatk`
`therefore` Line is parallel to the vector `(hati-hatj+2hatk)xx(3hati+hatj+hatk)=|{:(hati,,hatj,,hatk),(1,,-1,,2),(3,,1,,1):}|=-3hati+5hatj+4hatk`
Line passes through (1, 2, 3).
`therefore` Equation of line is `vecr=(hati+2hatj+3hatk)+lamda(-3hati+5hatj+4hatk)`
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