Final answer:
The vector equation of a line that passes through the point P=(-5,-2,-9) with the same direction as the given line L is r(t) = -5i - 2j - 9k + t(-1i + 3j + 2k).
Step-by-step explanation:
The student is asking for the vector equation of a line that passes through a specific point, given the parametric equations of another line. To find the vector equation for a new line that passes through the point P=(-5,-2,-9), we can use the direction vector from the original line L, which can be obtained from the coefficients of the parameter t in the given parametric equations. The direction vector for L is (−1, 3, 2), so any line parallel to L including the one through P will have the same direction vector.
The vector equation of a line can be expressed as r(t) = r0 + tv, where r0 is the position vector of a point on the line (in this case, point P), t is the parameter (which need not be the same as the one for line L), and v is the direction vector of the line. Therefore, the vector equation of the line through P with the direction vector −1, 3, 2) is:
r(t) = −5i − 2j − 9k + t(−1i + 3j + 2k)