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1. <x = 0 + 2t, y = 15 – 3t, z = -6 + 7t> is the vector parametric equation of the line parallel to the given one and that passes through the point (0, 15, −6).

   According to the statement

   we have to find that the vector equation with the help of the given line equation and the points.

   So, For this purpose, we know that the

   A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.

   And the given points is (0, 15, −6) and the lines

   x = −1 + 2t,

   y = 6 − 3t,

   z = 3 + 7t

   From these equation of lines:

   Let a = -1 and b = 6 and c = 3

   Then

   Now, if we want this line to pass through the point (0, 15, -6), then we can replace the correspondent values in the constant term for each equation:

   So, Put it in the given equations then

   x = 0 + 2t and y = 15 – 3t and z = -6 + 7t

   and the vector equation become

   <x = 0 + 2t, y = 15 – 3t, z = -6 + 7t>

   So, <x = 0 + 2t, y = 15 – 3t, z = -6 + 7t> is the vector parametric equation of the line parallel to the given one and that passes through the point (0, 15, −6).

   Learn more about vector parametric equation here

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