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Question
Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).
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Solution
Let the line passing through the points, P (3, −2, −5) and Q (3, −2, 6), be PQ.
Since PQ passes through P (3, −2, −5), its position vector is given by,
`veca = 3hati -2hatj - 5hatk`
The direction ratios of PQ are given by,
(3 − 3) = 0, (−2 + 2) = 0, (6 + 5) = 11
The equation of the vector in the direction of PQ is
`vecb = 0.hati - 0.hatj + 11hatk = 11hatk`
The equation of PQ in vector form is given by, `vecr = veca + lambdavecb`, `lambda in R`
`=>vecr = (3hati - 2hatj - 5hatk) + 11lambdahatk`
The equation of PQ in Cartesian form is
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