The line `vecr= veca + lambda vecb` will not meet the plane
`vecr cdot n =q,` if

Options

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

Answer

To determine the conditions under which 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 can follow these steps:

### Step 1: Understand the conditions for a line not meeting a plane
A line will not meet a plane if:
1. The line is parallel to the plane.
2. Any point on the line does not lie on the plane.

### Step 2: Condition for the line to be parallel to the plane
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:
\[ \vec{b} \cdot \vec{n} = 0 \]
This means that the dot product of the direction vector of the line and the normal vector of the plane must equal zero.

### Step 3: Condition for a point on the line not lying on the plane
Next, we need to ensure that any point on the line does not lie on the plane. A point on the line can be represented by \(\vec{r} = \vec{a}\) when \(\lambda = 0\). For this point to not lie on the plane, it must satisfy:
\[ \vec{a} \cdot \vec{n} \neq q \]
This means that the dot product of the position vector \(\vec{a}\) and the normal vector \(\vec{n}\) must not equal the constant \(q\).

### Step 4: Combine the conditions
For the line \(\vec{r} = \vec{a} + \lambda \vec{b}\) to not meet the plane \(\vec{r} \cdot \vec{n} = q\), both conditions must be satisfied:
1. \(\vec{b} \cdot \vec{n} = 0\) (the line is parallel to the plane)
2. \(\vec{a} \cdot \vec{n} \neq q\) (a point on the line does not lie on the plane)

### Conclusion
Thus, the line will not meet the plane if:
- \(\vec{b} \cdot \vec{n} = 0\)
- \(\vec{a} \cdot \vec{n} \neq q\)