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Question
Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines.
`vecr=(8hati-19hatj+10hatk)+lambda(3hati-16hatj+7hatk) " and "vecr=(15hati+29hatj+5hatk)+mu(3hati+8hatj-5hatk)`
Solution
The equations of the given lines are
`vecr=(8hati-19hatj+10hatk)+lambda(3hati-16hatj+7hatk) " ...(1)"`
`vecr=(15hati+29hatj+5hatk)+mu(3hati+8hatj-5hatk) " .....(2)"`
Normal parallel to (1) is `vec(n_1)=3hati-16hatj+7hatk`
Normal parallel to (2) is `vec(n_2)=3hati+8hatj-5hatk`
The required line is perpendicular to the given lines. So, the normal n⃗ parallel to the required line is perpendicular to `vec(n_1) " and "vec(n_2)`
`:.vecn=vec(n_1)xxvec(n_2)=|(hati, hatj,hatk),(3,-16,7),(3,8,-5)|=24hati+36hatj+72hatk`
Thus, the vector equation of the required line is
`vecr=(hati+2hatj-4hatk)+gamma(24hati+36hatj+72hatk)`
`=>vecr=(hati+2hatj-4hatk)+k(2hati+3hatj+6hatk) " (Where k=12γ)"`
Also, the Cartesian equation of the required line is
`(x-1)/2=(y-2)/3=(z+4)/6`
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