The Cartesian equation of the plane `vecr. (hati + hatj + hatk) =2 is "_______."`

Answer

To find the Cartesian equation of the plane given by the vector equation \(\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 2\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Vector Equation**: The vector equation \(\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 2\) represents a plane in three-dimensional space. Here, \(\vec{r}\) is the position vector of any point on the plane.

2. **Express \(\vec{r}\)**: The position vector \(\vec{r}\) can be expressed in terms of its components as:
\[ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \]
where \(x\), \(y\), and \(z\) are the coordinates of a point in the plane.

3. **Substitute \(\vec{r}\) into the Dot Product**: Substitute \(\vec{r}\) into the dot product:
\[ (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) = 2 \]

4. **Calculate the Dot Product**: Now, calculate the dot product:
\[ x\hat{i} \cdot \hat{i} + y\hat{j} \cdot \hat{j} + z\hat{k} \cdot \hat{k} + x\hat{i} \cdot \hat{j} + x\hat{i} \cdot \hat{k} + y\hat{j} \cdot \hat{i} + y\hat{j} \cdot \hat{k} + z\hat{k} \cdot \hat{i} + z\hat{k} \cdot \hat{j} \]
Since the dot product of different unit vectors is zero, we only consider the terms where the unit vectors are the same:
\[ = x + y + z \]

5. **Set the Equation Equal to 2**: Now, we set the result equal to 2:
\[ x + y + z = 2 \]

6. **Final Cartesian Equation**: Thus, the Cartesian equation of the plane is:
\[ x + y + z = 2 \]

### Final Answer:
The Cartesian equation of the plane is \(x + y + z = 2\).