Final answer:
The student requested parametric equations for a line given a point and a direction vector. The provided parametric equations are based on the given point P(3, -4, -1) and the direction vector i + 3j + k.
Step-by-step explanation:
The student asks for parametric equations for a line. To find these equations, we need a point through which the line passes and a direction vector. The student provides us with the point P(3, -4, -1) and the direction vector i + 3j + k. Since the line is parallel to this vector, our parametric equations will use its components as coefficients for the parameter t.
The general form of parametric equations for a line in three dimensions is:
- x = x_0 + at
- y = y_0 + bt
- z = z_0 + ct
Where (x_0, y_0, z_0) is a point on the line, and (a, b, c) is the direction vector components. For this example:
- x = 3 + 1⋅t
- y = -4 + 3⋅t
- z = -1 + 1⋅t
These are the required parametric equations for the line requested by the student.