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.
