(a) To obtain the vector equation of the line passing through the points (-4,-6,1) and (-2,0,-3), we need to first calculate the direction vector. The direction vector is obtained by subtracting the coordinates of the first point from the second. Upon performing this step, we find the direction vector to be (2, 6, -4).
We can represent any point on the line as a position vector from the origin, r = a + tb, where a is a point on the line, b is the direction vector, and t is a scalar. We know that the point (-4,-6,1) is on the line, and hence, can choose this as the position vector. Therefore, the vector equation of the line is given by r = position_vector + t * direction_vector.
(b) In order to determine if the line described in part (a) is parallel to the line passing through the points (10,18,4) and (5,3,14), we need to calculate the direction vector of the second line. Subtraction of the coordinates gives us the direction vector for the second line as (-5, -15, 10).
Two lines are parallel if their direction vectors are proportional. We can check this by dividing each component of the first line's direction vector by the corresponding component of the second line's direction vector. If these division results are equal for each component, then the direction vectors (and thus, the lines) are indeed parallel. Upon performing this operation, we find that all three division results are indeed equal. Hence, the two lines are parallel.
(c) The reasoning in the previous part is sufficient justification. Since the direction vectors of the two lines are proportional (as seen by equal division results), the two lines are parallel.
(d) A line intersects the xy plane when the z-component of the line equation equals zero. To determine the point at which this occurs, we have to solve for t in the equation given by the z-component of the line (equated to zero): point1[2] + t * direction_vector[2] = 0. Upon solving this, we find that t = 0.25.
We then substitute this value of t back into the line equation to find the position vector of the intersection point. After performing the calculations, we find that the intersection point is (-3.5, -4.5, 0).