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Numerade Educator

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Problem 42 Medium Difficulty

Let $ v = 5j $ and let $ u $ be a vector with length 3 that starts at the origin and rotates in the $ xy $-plane. Find the maximum and minimum values of the length of the vector $ u \times v $. In what direction does $ u \times v $ point?

Answer

$\max =15$
$\min =0$

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Video Transcript

Let's talk about this question. Were given that we visible to five J. and you be a factor of land three. That starts at the original rotates and life weapon. All right. So we're not given positive direction of the factor we all we are given as the direction. Or let's say the factor we you is something V. Is something which we argument. So we is five years. So five J. Is definitely like this. And there is a factor uh you which is an experiment but it rotates it is a length three. Its magnitude is three but it rotates and the X. Factor. So we don't know the direction of the of the U. All right. So in the maximum and then we'll have the length of the of the battery you cross. Which we've got to find the magnitude or as in the absolute value of you cross We've actors. That's got to be absolute value of U. Times absolute value of three times sine of the angle which is between them. Alright. So clearly for the maximum for the maximum the magnitude of you cross we signed Pita should be maxim. And we know that the value of science it is maximum when it is equal to one and this equal to one means the angle between them must be 90° must be 90°. That's why the maximum and minimum length and we are just they're just asking us the length. So the length maximum is going to be when we replace scientific by one. So the maximum will be absolute value of you. An absolute value of we an absolute value of you is already given to us the length of the U. S. Three. Right? So we're gonna replace that by three and V. S. Five G's. A. And the the magnitude of five G. Is just five. So the maximum value is 15. This is the maximum value of the cross product of you. And we and part B asks about the minimum values. You got to do the same logic. But the thing with minimum is so it's a minimum minimum for the minimum magnitude uh definitely uh uh This and this also has an absolute value sign over here for the minimum magnitude. Uh We have to make scientific to as low as possible but definitely it should be positive. So although we can make scientific as minus one but that won't really make any sense because minus absolute value is one and uh minus one of two values one. So it will again be maximum. So what we have to do is we have to make scientific as zero. It means that the angle between them angle between them has to be zero degrees or 1 80 degrees angle between the two actors. And hence the minimum minimum length of the cross product of these. It's got to be absolute value of U. Times absolute value of V. Times zero because Science 00. And this is just zero. Uh They're also asking that what is the direction in which you cross we point. So remember whenever we have across product, then the output actually points in the direction which is are you gonna to both these factors? So the cross product is gonna point. So you cross we is gonna point either towards plus J. Or minus your access. So that's what we're gonna right here. That you cross. We knew cross we will point towards plus. Okay, or minus Cape. Thank you.