1.

Find the vector equation of the line passingthrough the point (1,2,3) and parallel to the planes ` vec rdot( hat i- hat j+2 hat k)=5`and ` vec rdot(3 hat i+ hat j+ hat k)=6.`

Answer» `P_1=vecr*(hati-hatj+2hatk)=5`
`vecn_1=hati-hatj+2hatk`
`P_2=vecr*(3hati+hatj+hatk)=6`
`vecn=3hati+hatj+hatk`
`l=vecr=veca+lambdavecb`
`vecb=vecn_1*vecn_2`
`=i(-1-2)-j(1-6)+k(1-(-3))`
`vecb=-3hati+5hatj+4hatk`
`vecr=veca+lambdavecb`
`vecr=(hati+2hatj+3hatk)+lambda(-3hati+5hatj+4hatk)`
`vecr=(1-3lambda)hati+(2+5lambda)hatj+(3+4lambda)hatk`.


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