Find the cartesian equation of the plane whose vector equation is `vecr.(3hati-5hatj+7hatk)+8=0`

Options

A. 3x + 5y + 72+8=0
B. 3x - 5y +72-8=0
C. 3x - 5y - 72 +8 = 0
D. 3x - 5y +72+8=0

Answer

To find the Cartesian equation of the plane given its vector equation, we can follow these steps:

### Step 1: Understand the Vector Equation
The vector equation of the plane is given as:
\[ \vec{r} \cdot (3\hat{i} - 5\hat{j} + 7\hat{k}) + 8 = 0 \]
Here, \(\vec{r}\) represents a general point in the plane, which can be expressed as:
\[ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \]

### Step 2: Substitute \(\vec{r}\) into the Equation
Substituting \(\vec{r}\) into the vector equation, we have:
\[ (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (3\hat{i} - 5\hat{j} + 7\hat{k}) + 8 = 0 \]

### Step 3: Calculate the Dot Product
Now, we calculate the dot product:
\[ x \cdot 3 + y \cdot (-5) + z \cdot 7 + 8 = 0 \]
This simplifies to:
\[ 3x - 5y + 7z + 8 = 0 \]

### Step 4: Rearrange to Get the Cartesian Equation
To express this in the standard Cartesian form, we rearrange the equation:
\[ 3x - 5y + 7z = -8 \]

### Final Result
Thus, the Cartesian equation of the plane is:
\[ 3x - 5y + 7z + 8 = 0 \]

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