To answer the question, calculate the intersection point on line L using the three-dimensional Pythagorean theorem and from there determine a direction vector using P and this intersection point. Apply vector addition and scalar multiplication to write the vector equation for the desired line that goes through P and intersects L at the specified distance from Q.
To find the vector equation for the line that passes through the point P=(-10,-7,2) and intersects line L at a point that is a distance of 5 from point Q=(1,3,0), we must utilize the concepts of vector addition and scalar multiplication. First, determine the point of intersection on line L that is 5 units from Q. Using a three-dimensional version of the Pythagorean theorem (r² = x² + y² + z²), we can calculate distances and compare them to the required 5 units.
Once the intersection point on L is found, compute the direction vector from P to this point. The vector equation of the desired line will be x = P + tV, where P is the position vector of point P, V is the direction vector from P to the intersection point on L, and t is a scalar parameter.
In this case, a detailed diagram and specific calculations based on given parametric equations and distances would normally be required. However, due to the hypothetical nature of the information provided, a more precise solution isn't feasible without additional specifics or clarifications. In conclusion, with the provided parametric equations and suitable trigonometric applications, one could find the exact vector equation for the required line.