Q. 13 - Calculus

Question

What point is symmetric to the point \((3, −7, −4)\) with respect to the plane \(z = 1\)? 

Step-by-Step Solution

Answer

The point that is symmetric to the point \((3, −7, −4)\) with respect to the plane\( z = 1\)  is \((3,-7, 6)\)

Step 1. Given information

The given point is \((3, −7, −4)\).

Here, both the y and z coordinates are negative and the x-coordinate is positive. Thus, the above point lies in the fourth octant.

Step 2. Finding the required symmetric point.

As the given point lies in the fourth octant, its symmetric point with respect to the plane \(z=1\) will lie at the right of this plane (third octant) where the z-coordinate is positive.

Here given plane is \(z=1\)

Distance between the z-coordinate of the given point i.e -4 and plane z=1 is given as:

\(z = 1-(-4)\)

    = 5

Therefore, the new z-coordinate in the third octant is :
z = 1+5 =6

Therefore, The point that is symmetric to the point \((3, −7, −4)\) with respect to the plane \(z = 1\)  is \((3,-7, 6)\).