Video Transcript
Determine if the planes two, three, four dot 𝐫 equals 14 and four, six, eight dot 𝐫 equals 34 are parallel or perpendicular.
We see that the two planes given to us are expressed in what’s called vector form. Written this way, the vector that is dotted with the vector 𝐫 is normal to that plane. So, if we say that this first equation refers to plane one while the second refers to plane two, then by looking at plane one’s equation, we can say that a vector normal to this plane — we’ll call it 𝐧 one — has components two, three, four. Likewise, for plane two, a vector normal to it, we’ll call 𝐧 two, has components four, six, eight. Using these normal vectors, we can test whether planes one and two are parallel or perpendicular.
In general, if two planes with normal vectors 𝐧 one and 𝐧 two, respectively, are parallel, then we can find some constant 𝐶 by which we multiply the one normal vector to be equal to the other. If we can’t do this, that is if there is no such constant 𝐶 by which we can make these two normal vectors equal, then the planes aren’t parallel. In that case, they may be perpendicular. And the condition for this is that the dot product of the two normal vectors is zero.
Let’s now test planes one and two for these relationships, and we’ll start by seeing whether they’re parallel. One way to do this is to start by looking at the 𝑥-components involved. Since we’re searching for a constant 𝐶 by which we can multiply one of the normal vectors to equal the other, we can solve for the value of 𝐶 that makes this equation true. We know that if 𝐶 is one-half, then one-half times four is two, and the equation holds. In order for these two planes to be parallel, though, we need to be able to use the same constant for the 𝑦- and 𝑧-components.
Looking now at the 𝑦-components, let’s see if we can do that. We want to see if three is equal to our constant 𝐶 times six. Because 𝐶 is equal to one-half and one-half times six is three, this relationship does hold true, just like our relationship for the 𝑥-values. Lastly, then we’ll check the 𝑧-values of these vectors. That is, we’re seeing if four is equal to 𝐶 times eight, where, again, 𝐶, is one-half. One-half times eight is four.
And therefore, there does exist a constant value by which we can multiply one of our normal vectors to make it equal to the other. And that tells us that these planes are parallel, and that’s our answer to how these two planes relate.
