Let's analyze and solve this task on the relative positions between a line $L$ and a plane $P$, based on the conditions described.

The given problem asks about the relationships between a line $L$ and a plane $P$, depending on the alignment of their vectors:

1. $\vec{n}$: The normal vector of the plane $P$.
2. $\vec{q}$: The direction vector of the line $L$.

Now, let's evaluate the conditions:

Analysis of Each Option:

a) $L \parallel P$, if the normal vector $\vec{n}$ of $P$ and the direction vector $\vec{q}$ of $L$ are collinear.

• If $\vec{n}$ and $\vec{q}$ are collinear, the line $L$ lies entirely in the direction of the normal vector to the plane. This implies that the line $L$ cannot be parallel to the plane because it would either intersect or be perpendicular to it.
• This option is incorrect.

b) $L \perp P$, if the normal vector $\vec{n}$ of $P$ and the direction vector $\vec{q}$ of $L$ are collinear.

• If $\vec{n}$ and $\vec{q}$ are collinear, the line $L$ is aligned with the normal to the plane. This makes the line $L$ perpendicular to the plane.
• This option is correct.

c) $L \parallel P$, if the normal vector $\vec{n}$ of $P$ and the direction vector $\vec{q}$ of $L$ are orthogonal.

• If $\vec{n}$ and $\vec{q}$ are orthogonal, the line $L$ lies in a direction parallel to the plane since it cannot point "out of" the plane.
• This option is correct.

d) $L \perp P$, if the normal vector $\vec{n}$ of $P$ and the direction vector $\vec{q}$ of $L$ are orthogonal.

• If $\vec{n}$ and $\vec{q}$ are orthogonal, the line $L$ is not perpendicular to the plane. Instead, it is parallel or lies inside the plane.
• This option is incorrect.

Correct Answers:

• b: $L \perp P$ if $\vec{n}$ and $\vec{q}$ are collinear.
• c: $L \parallel P$ if $\vec{n}$ and $\vec{q}$ are orthogonal.

Do you want a detailed vector explanation for each case?

Related Questions:

1. What is the condition for a line to lie within a plane instead of being parallel to it?
2. How can you calculate if two vectors are orthogonal?
3. Can a line be both parallel and perpendicular to a plane simultaneously?
4. How does the orientation of the normal vector define the geometry of the plane?
5. How do you determine the angle between a line and a plane mathematically?

Tip:

When determining the relationship between a line and a plane, always check if the line's direction vector lies in the plane or aligns with the normal vector.