Find the equation ot the plane through the point p(5, 3, -1) perpendicular to the line in space
whose symmetric equations are `(x-2)/3=(y+1)/4=(z-1)/-2`. (Vector form)

Answer

To find the equation of the plane that passes through the point \( P(5, 3, -1) \) and is perpendicular to the line given by the symmetric equations

\[ \frac{x-2}{3} = \frac{y+1}{4} = \frac{z-1}{-2}, \]

we will follow these steps:

### Step 1: Identify the direction ratios of the line
The symmetric equations can be rewritten to identify the direction ratios. From the equations, we can see that the direction ratios of the line are \( (3, 4, -2) \).

### Step 2: Identify the normal vector of the plane
Since the plane is perpendicular to the line, the direction ratios of the line will serve as the normal vector \( \mathbf{n} \) of the plane. Thus, we have:

\[ \mathbf{n} = 3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}. \]

### Step 3: Write the position vector of the point
The position vector \( \mathbf{a} \) of the point \( P(5, 3, -1) \) is given by:

\[ \mathbf{a} = 5\mathbf{i} + 3\mathbf{j} - 1\mathbf{k}. \]

### Step 4: Use the vector form of the plane equation
The equation of the plane in vector form can be expressed as:

\[ \mathbf{r} - \mathbf{a} \cdot \mathbf{n} = 0, \]

where \( \mathbf{r} \) is the position vector of any point on the plane.

### Step 5: Substitute the known values into the equation
Substituting \( \mathbf{a} \) and \( \mathbf{n} \) into the equation, we have:

\[ \mathbf{r} - (5\mathbf{i} + 3\mathbf{j} - 1\mathbf{k}) \cdot (3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}) = 0. \]

### Step 6: Calculate the dot product
Now, we need to compute the dot product \( \mathbf{a} \cdot \mathbf{n} \):

\[ \mathbf{a} \cdot \mathbf{n} = (5)(3) + (3)(4) + (-1)(-2) = 15 + 12 + 2 = 29. \]

### Step 7: Write the final equation of the plane
Thus, the equation of the plane becomes:

\[ \mathbf{r} \cdot (3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}) = 29. \]

### Final Answer
The equation of the plane in vector form is:

\[ \mathbf{r} \cdot (3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}) = 29. \]

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