Question

The equation of a plane containing the point \( (1, -1, 1) \) and parallel to the plane \[ 2x + 3y - 4z = 17 \text{ is} \]

• A. \( \mathbf{r} \cdot (2\hat{i} + 3\hat{j} - 4\hat{k}) = -5 \)
• B. \( \mathbf{r} \cdot (2\hat{i} + 3\hat{j} - 4\hat{k}) = -15 \)
• C. \( \mathbf{r} \cdot (4\hat{i} + 3\hat{j} - 4\hat{k}) = -3 \)
• D. \( \mathbf{r} \cdot (3\hat{i} + 4\hat{j} - 2\hat{k}) = -3 \)

Hint:

For a plane parallel to another, use the same normal vector and substitute the given point to find the equation of the new plane.

Answer 1

The Correct Option is A

Step 1: Equation of a parallel plane.
The given plane has the equation \( 2x + 3y - 4z = 17 \), and we are asked to find the equation of a plane parallel to this one and passing through the point \( (1, -1, 1) \).
Step 2: Use the normal vector.
The normal vector to the plane is \( \mathbf{n} = (2, 3, -4) \), as it corresponds to the coefficients of \( x \), \( y \), and \( z \) in the equation of the plane.
Step 3: Substitute the point into the equation.
The equation of the plane passing through the point \( (1, -1, 1) \) and parallel to the given plane is of the form: \[ \mathbf{r} \cdot (2\hat{i} + 3\hat{j} - 4\hat{k}) = D, \] where \( D \) is a constant. Substituting the point \( (1, -1, 1) \) into the equation, we get: \[ (1, -1, 1) \cdot (2, 3, -4) = 2(1) + 3(-1) - 4(1) = -5. \] Thus, the equation of the plane is: \[ \mathbf{r} \cdot (2\hat{i} + 3\hat{j} - 4\hat{k}) = -5. \]
Step 4: Conclusion.
Thus, the correct answer is option (A).