Animated Solution for Mathematics - Three Dimensional Geometry: Consider the three planes $P_1:3x+15y+21z=9$, $P_2:x-3y-z=5$, and $P_3:2x+10y+14z=5$. Then, which one of the following is true?

Imagine you are standing in a vast, empty room. You have three giant, invisible sheets of glass floating in the air. These are our planes: $P_1$, $P_2$, and $P_3$.

Your mission is to determine which of these sheets are perfectly parallel to each other. It is a classic problem in 3D geometry, and it is more intuitive than it looks.

Every plane has a unique orientation, and we describe this orientation using a normal vector, $\vec{n}$. Think of this vector as an arrow sticking straight out of the surface of the plane.

If two planes are parallel, their arrows must point in the exact same direction (or the exact opposite). Mathematically, this means the components of their normal vectors must be proportional.

If we have two planes $a_{1}x+b_{1}y+c_{1}z=d_1$ and $a_{2}x+b_{2}y+c_{2}z=d_2$, they are parallel if:

Let us look at our first plane, $P_1:3x+15y+21z=9$. The coefficients are $3,15,$ and $21$.

Notice that all these numbers are divisible by $3$. When we divide the entire equation by $3$, we get $x+5y+7z=3$.

Now, the normal vector ${\vec{n}}_1$ is clearly $(1,5,7)$. This is the 'DNA' of plane $P_1$.

Now, let us look at $P_3:2x+10y+14z=5$. Again, the coefficients $2,10,$ and $14$ are all even.

Let us divide by $2$ to simplify. We get:

The normal vector ${\vec{n}}_3$ is $(1,5,7)$.

Compare ${\vec{n}}_1=(1,5,7)$ and ${\vec{n}}_3=(1,5,7)$. They are identical! This is the smoking gun.

Because their normal vectors are the same, the planes $P_1$ and $P_3$ must be parallel. But wait—are they the same plane?

We check the constants. For $P_1$, the constant is $3$. For $P_3$, it is $\frac{5}{2}$.

Since $3 eq \frac{5}{2}$, these planes are distinct. They are like two parallel floors in a building—perfectly aligned but separated by a gap.

Finally, let us check the middle child, $P_2:x-3y-z=5$. Its normal vector ${\vec{n}}_2$ is $(1,-3,-1)$.

If we compare this to our parallel pair, the ratios $\frac{1}{1},\frac{-3}{5},$ and $\frac{-1}{7}$ are clearly not equal.

$P_2$ is not parallel to either of them; it is a rebel, slicing through the space at a completely different angle.

By simplifying the equations and identifying the normal vectors, we have solved the mystery. $P_1$ and $P_3$ are the parallel pair.

Geometry is not just about memorizing formulas; it is about visualizing the space and finding the hidden patterns. You have just mastered the art of identifying parallel planes—keep that confidence as you tackle the next challenge!