Question

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

Short answer

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

Step-by-step solution

Introduction.

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

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.

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 role="math" localid="1654146314091" $z=1$is

role="math" localid="1654146330025" $z=1-(-4)=5.$

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 $(3,-7,6).$