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 symmetric towards to the point $(3,-7,-4)$ about the plane $z=1$ is $(3,-7,6).$

Step1: Introduction.

If the $y$ and $x$ values of a mathematical formula can be swapped, it is symmetrical to the line $y = x.$

Step2: Given Information.

Consider the following statement: 

$(3,-7,-4)$

The goal is to find the point on the plane $z=1$ that is symmetric to this point.

Step3: Explanation.

The point $(3,-7,-4)$ is in the fourth octant because the $y$ and $z$ coordinates are negative and the $x$ coordinate is positive.

The third octant is the point that is symmetric about the plane $z=1.$

The $x$ and $z$ coordinates are both positive in the third octant, while the $y$ coordinate is negative.

The distance between this point and the plane $z=1$ is

$z=1-(-4)=5.$

As a consequence, the newly created $z-$coordinate is

$z=1+5.$

As a result, the point is symmetric in relation to the point $(3,-7,-4)$ about the plane $z=1$ is $(3,-7,6).$