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

If 𝚨 = 4𝐒 + 𝐣, 𝚩 = (π‘˜ βˆ’ 2)𝐒 + 2𝐣, and 𝚨 βˆ₯ 𝚩, then π‘˜ = οΌΏ.

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Video Transcript

If 𝚨 equals four 𝐒 plus 𝐣, 𝚩 equals π‘˜ minus two times 𝐒 plus two 𝐣, and 𝚨 is parallel to 𝚩, then π‘˜ 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 𝐢 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 π‘₯-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 π‘₯ over 𝐡 sub π‘₯ equals 𝐴 sub 𝑦 over 𝐡 sub 𝑦. We see that 𝐴 sub π‘₯ equals four, 𝐡 sub π‘₯ equals π‘˜ minus two, while 𝐴 sub 𝑦 equals one and 𝐡 sub 𝑦 is two. We now have an equation where everything in it is known except for the unknown π‘˜.

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

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