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Question
Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).
Solution
The required line passes through the origin. Therefore, its position vector is given by,
`veca = vec0` ...(1)
The direction ratios of the line through origin and (5, −2, 3) are
(5 − 0) = 5, (−2 − 0) = −2, (3 − 0) = 3
The line is parallel to the vector given by the equation, `vecb = 5hati - 2hatj + 3hatk`
The equation of the line in vector form through a point with position vector `veca` and parallel to `vecb` is, `vecr = veca + lambdavecb`, `lambda in R`
The equation of the line through the point (x1, y1, z1) and direction ratios a, b, c is given by `(x-x_1)/x = (y-y_1)/b = (z-z_1)/c`
Therefore, the equation of the required line in the Cartesian form is
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