Video Transcript
Suppose vector 𝐀 equals one, three, two; vector 𝐁 equals 𝑘, nine, 𝑚; vector 𝐂 equals 𝑘, 𝑚, 𝑘 plus 𝑚; and 𝐀 is parallel to 𝐁, find the magnitude of vector 𝐂.
Of these three vectors, we see that we’re given all the components of vector 𝐀 while vector 𝐁 has these two unknown components 𝑘 and 𝑚, and therefore all the components of vector 𝐂 which depend on 𝑘 and 𝑚 are unknown. We’re given this bit of information, though, that vectors 𝐀 and 𝐁 are parallel. In general, if we have two three-dimensional vectors — we’ll call them 𝐮 and 𝐯 — when these vectors are parallel, that means we can write an equation like this. This relation tells us that the vectors are equal to one another to within some constant multiple 𝐶. 𝐶 can be any nonzero value. And so long as some such value exists, that makes this equation true; the vectors 𝐮 and 𝐯 are parallel.
If we write out the components of these two general vectors 𝐮 and 𝐯, then we can see there’s a second way to express this condition of the vectors being parallel. If the vectors are parallel, we can say that the ratio of their 𝑥-values is equal to the ratio of their 𝑦-values and the ratio of their 𝑧-values. These are two mathematically equivalent ways of expressing that these vectors are parallel. For our given vectors 𝐀 and 𝐁, we’ll use this second form to express their parallel condition. The 𝑥-component of 𝐀 is one, while the 𝑥-component of 𝐁 is 𝑘. And since 𝐀 and 𝐁 are parallel, this must equal the 𝑦-component of 𝐀 divided by the 𝑦-component of 𝐁 and then, likewise, the 𝑧-component of 𝐀 divided by the 𝑧-component of 𝐁.
In this expression, we have two unknowns 𝑘 and 𝑚 and two independent equations. We can write that one over 𝑘 equals three over nine. This implies that nine is equal to three times 𝑘 or that 𝑘 equals three. Likewise, our oval in pink shows us that three over nine equals two over 𝑚. If we cross multiply by nine and 𝑚, we find that three 𝑚 equals two times nine, which means that three 𝑚 equals 18 or 𝑚 is equal to six. Now that we know the values of 𝑘 and 𝑚, we can write out the components of vector 𝐂. They’re equal to 𝑘 which is three, 𝑚 which is six, and 𝑘 plus 𝑚 which is nine.
Our goal is to solve for the magnitude of this vector; that involves computing the square root of the sum of the squares of the three components of vector 𝐂. Three squared is nine, six squared is 36, and nine squared is 81. The magnitude of 𝐂 then equals the square root of 126. We can note, though, that 126 is equal to nine times 14. And then, since the square root of nine is three, we can simplify this result to three times the square root of 14. This is the magnitude of vector 𝐂 which we saw is based on the components of vector 𝐁.
