Plane

Step 1: Set up the equation for the plane.
The equation of the plane is: We are given that the position vector of the point is , which corresponds to the point .

Step 2: Use the reflection formula.
To find the image of the point in the plane, we use the formula for the reflection of a point across a plane. If the point is reflected across a plane with normal vector , the image point is given by: Here, , and .

Step 3: Calculate the dot product.
The dot product is:

Step 4: Find the scalar for the reflection.
Now, calculate the scalar:

Step 5: Calculate the image vector.
Using the formula for reflection: Thus, the image of the point is , corresponding to option (C).

Step 1: Use the given point and vectors.
We are given the point and the vectors and , which define the direction of the plane. The general equation of a plane is: where is the position vector of any point on the plane and is the normal vector to the plane.

Step 2: Find the normal vector.
The normal vector is perpendicular to both and . Thus, we find by taking the cross product of and : Thus, .

Step 3: Equation of the plane.
The equation of the plane is: Simplifying the equation:

Step 4: Conclusion.
Thus, the equation of the plane is , corresponding to option (C).

Concept: Equation of plane through intersection of two planes:

Step 1: Form general equation.

Step 2: Substitute point :

Step 3: Substitute : Multiply by 3: Final Answer:

Concept: Check which point satisfies the plane equation.

Step 1: Substitute point .

Step 2: Substitute point : Final Answer:

Concept: Image of a point in a plane is found using reflection formula.

Step 1: Given point: Plane:

Step 2: Use reflection formula: Substitute: Final Answer:

Concept: A plane perpendicular to another plane has its normal vector perpendicular to the given plane’s normal.

Step 1: Normal of given plane.


Step 2: Direction vector of line joining given points.


Step 3: Normal of required plane.
Required plane contains line AB and is perpendicular to given plane. So its normal is perpendicular to both and :

Step 4: Equation of plane.
Using point :