The correct answer is C) x = t, y = 2t, z = 3t.
The correct answer is C) x = t, y = 2t, z = 3t. Here's why:
Parametric equations represent a line using a single parameter, typically denoted as `t`. These equations express the coordinates of any point on the line as functions of `t`.
For a line passing through a point `P(a, b, c)` and parallel to a direction
the parametric equations are:
where `t` varies across all real numbers.
Let's analyze each option:
* A) x = 2t, y = 3t, z = 4t: This represents a line with direction vector (2, 3, 4), but it doesn't specify the point it passes through.
* B) x = 3t, y = 4t, z = 5t: Similar to A, this defines a direction but lacks the point information.
* C) x = t, y = 2t, z = 3t: This perfectly matches the general form with `d = (1, 2, 3)` and an unspecified point on the line.
* D) x = 4t, y = 5t, z = 6t: This represents a line parallel to (4, 5, 6), but again, the point is missing.
Therefore, only option C provides valid parametric equations for a line passing through an unspecified point and parallel to a given vector.
Remember, when dealing with parametric equations, the key is to identify the direction vector and the general form of the equations. Once you have those, you can adjust the constants based on the specific point and direction you're given.