Final answer:
To find a unit normal to the plane, you can find two points in the plane using the given information. Then, subtract the coordinates of these two points to find two direction vectors. Finally, take the cross product of these direction vectors and normalize the result to find a unit normal.
Step-by-step explanation:
To find a unit normal to the plane, we need to find two direction vectors in the plane. Let's find two points in the plane using the given information:
For the point (1, -2, 1), we can take this as one point in the plane.
For the line x = 3t, y = 4t, z = 0, we can take the point (3, 4, 0) as another point in the plane.
Now, we can find the direction vectors by subtracting the coordinates of these two points.
The first direction vector is (3 - 1, 4 - (-2), 0 - 1) = (2, 6, -1).
The second direction vector is (3 - 1, 4 - (-2), 0 - 1) = (2, 6, -1).
To find a unit normal, we can take the cross product of these two direction vectors and then normalize the result.
The cross product is (2, 6, -1) × (2, 6, -1) = (12, -4, 16).
Normalizing the result, we get a unit normal (12/20, -4/20, 16/20) = (0.6, -0.2, 0.8).