The line `vecr= veca + lambda vecb` will not meet the plane
`vecr cdot vecn =q,` if a.`vecb.vecn=0,veca.vecn=q` b.`vecb.vecn" "ne0,veca.vecn" "ne q` c.`vecb.vecn=0,veca.vecn" "ne q` d.`vecb.vecn" "ne0,veca.vecn" "ne q`

Options

A. `vecb.vecn=0,veca.vecn=q`
B. `vecb.vecn" "ne0,veca.vecn" "ne q`
C. `vecb.vecn=0,veca.vecn" "ne q`
D. `vecb.vecn" "ne0,veca.vecn" "ne q`

Explanation

To determine when the line given by the equation \(\vec{r} = \vec{a} + \lambda \vec{b}\) does not meet the plane defined by the equation \(\vec{r} \cdot \vec{n} = q\), we need to analyze the conditions under which the line is parallel to the plane and does not intersect it.

### Step-by-Step Solution:

1. **Understanding the Line and Plane**:
- The line is represented as \(\vec{r} = \vec{a} + \lambda \vec{b}\), where \(\vec{a}\) is a point on the line and \(\vec{b}\) is the direction vector of the line.
- The plane is given by the equation \(\vec{r} \cdot \vec{n} = q\), where \(\vec{n}\) is the normal vector to the plane.

2. **Condition for Parallelism**:
- For the line to be parallel to the plane, the direction vector of the line \(\vec{b}\) must be perpendicular to the normal vector of the plane \(\vec{n}\). This can be expressed mathematically as:
\[ \vec{b} \cdot \vec{n} = 0 \]
- If this condition holds, the line does not intersect the plane.

3. **Condition for Not Passing Through the Plane**:
- Even if the line is parallel to the plane, it must also not pass through any point of the plane. This means that the point \(\vec{a}\) (which lies on the line) should not satisfy the plane's equation:
\[ \vec{a} \cdot \vec{n} \neq q \]

4. **Combining the Conditions**:
- Therefore, the line \(\vec{r} = \vec{a} + \lambda \vec{b}\) will not meet the plane if:
- \(\vec{b} \cdot \vec{n} = 0\) (line is parallel to the plane)
- \(\vec{a} \cdot \vec{n} \neq q\) (line does not pass through the plane)

5. **Identifying the Correct Option**:
- Based on the conditions derived:
- **Option A**: \(\vec{b} \cdot \vec{n} = 0\), \(\vec{a} \cdot \vec{n} = q\) (does not satisfy the second condition)
- **Option B**: \(\vec{b} \cdot \vec{n} \neq 0\), \(\vec{a} \cdot \vec{n} \neq q\) (does not satisfy the first condition)
- **Option C**: \(\vec{b} \cdot \vec{n} = 0\), \(\vec{a} \cdot \vec{n} \neq q\) (satisfies both conditions)
- **Option D**: \(\vec{b} \cdot \vec{n} \neq 0\), \(\vec{a} \cdot \vec{n} \neq q\) (does not satisfy the first condition)

- The correct option is **C**: \(\vec{b} \cdot \vec{n} = 0\) and \(\vec{a} \cdot \vec{n} \neq q\).