Question Video: Finding the Component of a Vector Parallel to a Known Vector Mathematics

Video Transcript

If 𝚨 equals four 𝐢 plus 𝐣, 𝚩 equals 𝑘 minus two times 𝐢 plus two 𝐣, and 𝚨 is parallel to 𝚩, then 𝑘 equals what.

Here we have these two two-dimensional vectors 𝚨 and 𝚩, and we’re told that they’re parallel to one another. Knowing this, we can recall that, in general, whenever we have two vectors, say we call them 𝐕 one and 𝐕 two, that are parallel, this means there exists some nonzero constant 𝐶 by which we can multiply 𝐕 two so it equals 𝐕 one. Another mathematically equivalent way of saying this is to write that the ratio of the 𝑥-component of 𝐕 one to 𝐕 two equals the 𝑦-component of 𝐕 one to 𝐕 two.

And we can apply this relationship to our two given vectors 𝚨 and 𝚩. Because 𝚨 and 𝚩 are parallel, we can say that 𝐴 sub 𝑥 over 𝐵 sub 𝑥 equals 𝐴 sub 𝑦 over 𝐵 sub 𝑦. We see that 𝐴 sub 𝑥 equals four, 𝐵 sub 𝑥 equals 𝑘 minus two, while 𝐴 sub 𝑦 equals one and 𝐵 sub 𝑦 is two. We now have an equation where everything in it is known except for the unknown 𝑘.

If we multiply both sides of this equation by 𝑘 minus two and multiply both sides by two, we find the result that four times two equals one times the quantity 𝑘 minus two or eight equals 𝑘 minus two. Adding two to both sides, we find that 𝑘 equals 10. That’s our answer. So we can say that if 𝚨 equals four 𝐢 plus 𝐣, 𝚩 equals 𝑘 minus two 𝐢 plus two 𝐣, and 𝚨 is parallel to 𝚩, then 𝑘 equals 10.