In space, how many lines can be drawn through R perpendicular to ?

Theorem 3-9 can be used which states that a perpendicular line to another line consist of a point that is positioned on outer area of another line. This theorem is true for plane but it is not true for space because the count of lines for space is infinite.

The mentioned point in theorem according to the question is R and the another line is $\overleftrightarrow{QR}$. According to the figure the line on which R is positioned is $\overleftrightarrow{AR}$that is having a perpendicular relation with $\overleftrightarrow{QR}$along with this the consideration of another lines will result in formation of an angle which is equal to $90°$so the count of lines having a perpendicular relation with the line $\overleftrightarrow{QR}$is infinite because it is a space not plane.

As the count of line perpendicular to $\overleftrightarrow{QR}$is infinite which results in that the required count is infinite.