Q.The plane parallel to $yz$-plane is perpendicular to ________.

The key idea is that a plane parallel to the $yz$-plane has a constant $x$-coordinate, so its normal vector is along the $x$-axis.

Reasoning:

1. The $yz$-plane is defined by $x = 0$, so its normal vector is $\hat{i}$ (the unit vector along the $x$-axis).
2. Any plane parallel to the $yz$-plane has the same normal direction, i.e., along the $x$-axis. …

Understanding 3D Coordinate Octants and Planes

In three-dimensional coordinate geometry, the coordinate planes divide space into eight octants. The three fundamental planes are:

• $yz$-plane: All points where $x = 0$. This plane contains the $y$ and $z$ axes.
• $zx$-plane: All points where $y = 0$.
• $xy$-plane: All points where $z = 0$.

A plane parallel to the $yz$-plane means it never intersects the $yz$-plane — it is shifted along the $x$-axis. Such a plane has the equation $x = k$, where $k$ is a constant. For example, $x = 3$ is a plane parallel to the $yz$-plane, located 3 units away along the $x$-axis.

Now, what does "perpendicular to" mean here? A plane is perpendicular to a line if the line is normal (perpendicular) to the plane. So we need to find which axis or line is perpendicular to a plane parallel to the $yz$-plane.

Step-by-Step Reasoning

1. Equation of a plane parallel to $yz$-plane

   Any plane parallel to the $yz$-plane has the form $x = c$, where $c$ is a constant. This is because the $yz$-plane itself is $x = 0$, and moving it parallel means only the $x$-coordinate changes.

2. Normal vector of the plane

   The normal vector to a plane $x = c$ is $\vec{n} = (1, 0, 0)$. Why? Because the plane's equation can be written as $1 \cdot x + 0 \cdot y + 0 \cdot z = c$, so the coefficients of $x, y, z$ give the normal vector. This vector points along the positive $x$-axis.

3. Perpendicularity condition …