Final answer:
The student is seeking the vector equation of a line that passes through the intersection of two other lines. The equation follows the general form r = r_0 + μv, but without additional information on vectors a, b, and c, it's not possible to definitively select the correct option from the ones provided.
Step-by-step explanation:
The student is asking for the vector equation of a line in three-dimensional space that passes through a specific point. This point is the intersection of two lines, which are determined by the given points a+2b–5c, –a–2b–3c, and –4c, 6a–4b+4c. To find the required vector equation, we use the concept that lines in three-dimensional space can be represented by parametric equations using a point and a direction vector. The vector equation of a line is typically expressed as r = r_0 + μv, where r is the position vector of any point on the line, r_0 is the position vector of a fixed point on the line, v is the direction vector, and μ is the scalar parameter.
Examining the given options and cross-referencing with the data we have, none of the given sets of vectors appears to accurately represent the scenario without additional information regarding the specific nature of the given vectors a, b, and c, as well as how they relate to the points given. Thus, without a clearer context or direct calculations, we cannot definitively choose an option.