Find the cartesian equation of the plane `vec(r). (2 hati + 3 hatj - 4 hatk) = 1`.

Answer

To find the Cartesian equation of the plane given by the vector equation \(\vec{r} \cdot (2 \hat{i} + 3 \hat{j} - 4 \hat{k}) = 1\), we can follow these steps:

### Step 1: Understand the given vector equation
The vector equation of the plane is given in the form \(\vec{r} \cdot \vec{n} = p\), where \(\vec{r}\) is the position vector of any point on the plane, \(\vec{n}\) is the normal vector to the plane, and \(p\) is a constant.

### Step 2: Identify the components
In our case, we have:
- \(\vec{n} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}\)
- \(p = 1\)

### Step 3: Write the position vector
The position vector \(\vec{r}\) can be expressed in terms of its components as:
\[ \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} \]

### Step 4: Substitute into the equation
Now we substitute \(\vec{r}\) and \(\vec{n}\) into the equation:
\[ (x \hat{i} + y \hat{j} + z \hat{k}) \cdot (2 \hat{i} + 3 \hat{j} - 4 \hat{k}) = 1 \]

### Step 5: Compute the dot product
Calculating the dot product:
\[ x \cdot 2 + y \cdot 3 + z \cdot (-4) = 1 \]
This simplifies to:
\[ 2x + 3y - 4z = 1 \]

### Step 6: Write the Cartesian equation
Thus, the Cartesian equation of the plane is:
\[ 2x + 3y - 4z = 1 \]

### Summary of the solution
The Cartesian equation of the plane is \(2x + 3y - 4z = 1\).

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