Question Video: Finding Unknown Terms in the Equations of Two Parallel Planes Mathematics

Video Transcript

Given that the plane 𝐾𝑧 plus two 𝑥 plus three 𝑦 equals negative four is parallel to the plane 𝐿𝑦 minus two 𝑥 minus two 𝑧 equals three, find the values of 𝐾 and 𝐿.

Okay, so in this exercise we have these two planes given by these equations. If we write them in standard form, the first one is two 𝑥 plus three 𝑦 plus 𝐾𝑧, where 𝐾 is some constant value, is all equal to negative four. And the second is negative two 𝑥 plus 𝐿𝑦, where 𝐿 is some unknown constant, minus two times 𝑧 is equal to three. Because these equations are now written in standard form, we can pick out the components of the normal vectors to each plane. The normal vector of the first plane will have an 𝑥-component of two, a 𝑦-component of three, and a 𝑧-component of 𝐾. We’ll call this vector 𝐧 one and write it out this way. And then for the second plane, its normal vector will have an 𝑥-component of negative two, a 𝑦-component of 𝐿, and a 𝑧-component of negative two. And we’ll call this vector 𝐧 two.

Our problem statement tells us that these two planes are parallel to one another. This fact can point us to recalling the mathematical condition for two planes to be parallel. Two parallel planes with normal vectors 𝐧 one and 𝐧 two have their components related by a constant; we’ve called it 𝐶. This constant could be positive or negative, but whatever its sign, it means that there is a consistent ratio between the components of these two normal vectors. Knowing this, let’s look at our two normal vectors 𝐧 one and 𝐧 two and see what this constant of proportionality 𝐶 might be.

Comparing the 𝑥-components of these vectors, we see the 𝑥-component of 𝐧 one is two and that of 𝐧 two is negative two. We know that two is equal to negative one times negative two. And in fact, this is the only value by which we can multiply negative two to yield positive two which tells us that 𝐶, in this case, is negative one. And this fact is the key for solving for the values of 𝐾 and 𝐿 in our normal vector equations because just as we can write this equation out for the 𝑥-components of our normal vectors, so we can write it out for the 𝑦- and 𝑧-components.

If we substitute negative one in for 𝐶 in both of these equations, then we see that three is equal to negative one times 𝐿, telling us that 𝐿 is equal to negative three. And since 𝐾 is equal to negative one times negative two, that means 𝐾 is equal to two. So we’ve used the fact that our two planes are parallel to solve for the unknown parts of their normal vectors. 𝐿 is equal to negative three, and 𝐾 is equal to positive two.