💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

Numerade Educator

Like

Report

Problem 44 Hard Difficulty

(a) Find all vectors $ v $ such that
$$ \langle 1, 2, 1 \rangle \times v = \langle 3, 1, -5 \rangle $$
(b) Explain why there is no vector $ v $ such that
$$ \langle 1, 2, 1 \rangle \times v = \langle 3, 1, 5 \rangle $$

Answer

a. $\mathbf{v}=<x, 2 x-5, x-1>$
b. no possible solutions

More Answers

Discussion

You must be signed in to discuss.

Video Transcript

Welcome back to another cross products problem. This time we're trying to find all vectors V. That's right, that is X. Y. Z. Such that 1- one. Cross v equals 3. -5. Let's go ahead and write these in our matrix here. 1 to 1. And we said V was X Y Z. So we can calculate the cross product using the method in our textbook. Remember we eliminate the first column of our matrix and look at two Z minus one Y to z minus. Why? All multiplied by high minus. Then we ignore the second column. Look at one Z -1 X. The minus X. Time's jay. And lastly we'll ignore the third column And look at one way and a few X. I -2 x. Okay. Simplifying this a little bit gives us the vector to Z minus. Why negative Z minus X. That's x minus C. And why -2 X. Which should give us the same thing as the answer that we want. 31 negative five. This means that we have three linear equations here to the -Y equals three X -Z equals one. And Why -2? X equals negative five. So we have three equations with three unknowns. There's a couple different ways we can proceed here. But since we want something in the form X, Y. Z. Let's go ahead and write Y and Z. In terms of X. It's using these two equations. We know that Z is equal to x minus one, adding a Zeon, subtracting one from both sides. And we know that why is equal to two x minus five. One of the things we can do is check what's going on up here, You know that two times E minus? Why should equal three. And sure enough, two x -2 minus two, x minus five. These cancel -2 -5, no negative two plus five. There we go, negative two plus five Equals three. So these are consistent and we are looking at vectors in the form x x is just equal to x. Why is equal to two x -5 and Z is equal to x minus what? So for any value of x We can write v as X 2, -5, X -1. And it will satisfy this equation up here. Now what if we change things around a little bit? What if we want 1-1? Cross v To equal 3. 1 positive five. How would that work? The same idea? We would still want uh to z minus Y equals three. We would still want x minus Z equals one. And this time we would want y minus two X To equal positive five. OK, easy enough. We still get Z Equals X -1 and this time we get y equals two X plus five. But what happens when we check this equation for consistency? Where we get to z That's two times X minus one minus Y. That's two X plus five. We want this to equal three. But what do we get? 2? X -2 minus two. X minus five. Well that's the same thing as two X -2 x. We're left with -7 equals three. And that doesn't work. That doesn't make any sense. Negative seven does not equal three. Therefore this equation has no solutions. There are no vectors V. Such that 1- one. Cross v equals 315. Thanks for watching.