Find the equation of a straight line in the plane `vecr.vecn=d` which is parallel to `vecr.vecn=d("where "vecn.vecb=0)`.

Options

A. `vecr=veca+((d-veca.vecn)/(n^(2)))vecn+lamdavecb`
B. `vecr=veca+((d-veca.vecn)/(n))vecn+lamdavecb`
C. `vecr=veca+((veca.vecn-d)/(n^(2)))vecn+lamdavecb`
D. `vecr=veca+((veca.vecn-d)/(n))vecn+lamdavecb`

Explanation

To find the equation of a straight line in the plane defined by the equation \(\vec{r} \cdot \vec{n} = d\) that is parallel to another line represented by \(\vec{r} = \vec{a} + \lambda \vec{b}\) (where \(\vec{n} \cdot \vec{b} = 0\)), we can follow these steps:

### Step 1: Identify the Plane and the Direction Vector
The plane is given by the equation \(\vec{r} \cdot \vec{n} = d\). The direction vector of the line we want to find is \(\vec{b}\), and it is stated that \(\vec{n} \cdot \vec{b} = 0\), meaning that \(\vec{b}\) is perpendicular to the normal vector \(\vec{n}\) of the plane.

### Step 2: Find the Foot of the Perpendicular
To find the foot of the perpendicular from the point with position vector \(\vec{a}\) to the plane, we can use the formula:
\[ \vec{P} = \vec{a} - \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|^2} \vec{n} \]
This gives us the position vector \(\vec{P}\) of the foot of the perpendicular.

### Step 3: Write the Equation of the Line
The line we are looking for will pass through the point \(\vec{P}\) and will be parallel to the direction vector \(\vec{b}\). Therefore, the equation of the line can be expressed as:
\[ \vec{r} = \vec{P} + \lambda \vec{b} \]
Substituting \(\vec{P}\) from Step 2, we have:
\[ \vec{r} = \left(\vec{a} - \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|^2} \vec{n}\right) + \lambda \vec{b} \]

### Step 4: Simplify the Equation
Combining the terms, we can express the equation of the line as:
\[ \vec{r} = \vec{a} + \lambda \vec{b} - \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|^2} \vec{n} \]

### Final Result
Thus, the equation of the straight line in the plane \(\vec{r} \cdot \vec{n} = d\) that is parallel to \(\vec{r} = \vec{a} + \lambda \vec{b}\) is:
\[ \vec{r} = \vec{a} + \lambda \vec{b} - \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|^2} \vec{n} \]