Final answer:
To find the equation of the line through point (-6,2,3) parallel to the given line, use the direction vector (2, 3, 1) from the given line's equations and apply it to the point to get the new line's parametric equations: x = -6 + 2t, y = 2 + 3t, z = 3 + t.
Step-by-step explanation:
The question is asking to find the equation of the line that passes through the point (-6,2,3) and is parallel to the line represented by the equations ⅓x = ⅓y = z - 1. To derive the desired line's equation, we first need to determine the direction vector of the given line, which also becomes the direction vector of the line we want to find, as parallel lines have the same direction vector. From the given equations, the direction vector is (2, 3, 1). Now, using the point (-6,2,3) and this direction vector, we can write the parametric equations of the new line as:
- x = -6 + 2t
- y = 2 + 3t
- z = 3 + t
where t is the parameter. Since all three parametric equations are linear and involve the same parameter t, they represent a straight line in three-dimensional space with the point (-6,2,3) and direction vector (2,3,1).