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Which of the following is true for all vectors U and V in R37A lu+v=lul+IvlB. |u-vl < lulllvlclu-M-llul-IvlD. lu+v?=lule+lv?E. lu-vl > lulllv...

Question

Which of the following is true for all vectors U and V in R37A lu+v=lul+IvlB. |u-vl < lulllvlclu-M-llul-IvlD. lu+v?=lule+lv?E. lu-vl > lulllv

Which of the following is true for all vectors U and V in R37 A lu+v=lul+Ivl B. |u-vl < lulllvl clu-M-llul-Ivl D. lu+v?=lule+lv? E. lu-vl > lulllv



Answers

Determine which of the following are defined for nonzero vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w} .$ Explain your reasoning. (a) $\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})$ (b) $(\mathbf{u} \cdot \mathbf{v}) \mathbf{w}$ (c) $\mathbf{u} \cdot \mathbf{v}+\mathbf{w}$ (d) $\|\mathbf{u}\| \cdot(\mathbf{v}+\mathbf{w})$

Hello, everybody. In this video, I'm gonna be showing you how to solve exercise 72 in Chapter 13 Section four of calculus early transit bill. Now, in this problem, they want us to verify whether or not the following statement is true or false. And if it is true, to give an explanation and if it's false to get the counter example in the statement that they provide is that for any three non zero vectors u V and W you dotted with the cross W is equal to W dotted with U cross V. Now this statement is true. And to show why this is true, I'm gonna use a result from exercise 62 which states that the quantity given in the right you don't be cross w can be found by taking the determinant of a three by three major whose top row has the components of you which old to Noah's U one U 2 83 The second row has the components of E V one. You 83 in his last year has a components of W W one. What were you two w three. And if you're not sure why, this is true. I recommend you go back and try it. That problem for yourself. But this will be an important result that will use for the remainder of the problem. Now the next step will take exploits a useful property of matrix. Determine it. And this property is that if we swap any two rows in this matrix, the determinant of that resulting matrix will be the negative of our original one. So let's go ahead and swamp rose one and three. We have W one w two W three in the top row. Now V one, V two and V three in the second row is before now. In the last year, we have you won. You too, You three. And let's go ahead and do this again. This time we'll switch the second and third rose because we get another negative out of doing this. That will cancel it with the original one. And so our result will have no negative sign in front of it anymore. Swapping those two rows, we have w one, w two and W three in the top row. As in the previous step, you won. You too. You three in the second row no. And then in the last year we now have you won you too, b three. But notice that upon doing this transformation we can use the same property from exercise 62 backwards. But now it's expressed as w as that is, a vector in the top row started with you cross V vectors in the second and third rose. This is exactly what we wanted to prove. And so, using this result for this quantity from a concise 62. To put this expression in terms of a matrix, determine it and then using the roof swapping property of matrix determinants, we can show that these two quantities are indeed equal to each other. And that's how you solve exercise 72.

Hello, everybody. In this video, I'm going to be showing you how to solve exercise 71 in Chapter 13 Section four of calculus early Transcendental. Now, in this problem that wants to prove whether or not the following statement is true with an explanation or false with a counter thing in the statement they give us is that for any to non zero vectors U and V U minus V cross with U plus V is equal to two times U cross V, and this statement is true to show that it is true. What we want to do is expand out the left hand side of this equation and simplify using various properties of the cross product the first property will use is the distributive property. Using this property can say you minus V crossed with U plus V is equal to you minus V cross with you plus you minus V crossing the using the distributive property to split the second term in the cross product originally. But now we can use the distributive property again to split up the terms U minus V. In each of these two cross products, we have as a result for this first term this is equal. You cross you minus V across you. And then from the second term, the ad you cross V minus V Cross V and now we can eliminate some of the simpler terms. Clearly, any vector that is cross with itself is equal to zero. So we can eliminate you. Cross you as well. It's V cross V and what we're left with is negative view cross you plus U cross V. And now we want to turn to the anti community of property of the cross product. Using this property, we can write that negative V cross you is equal to U cross V, and therefore this term turns into U cross V and then we have you cross me twice so we can see that this expression is equal two times you crusty And now the last property we want to use is the associative property. Using this property, we can put the to into any one of these two vectors and maintain the same cross party so we can just put it into the first factor. You and we have that ultimately, our original expression is equal to to you ferocity. And this is exactly what we wanted to prove before. And so that's how you solve exercise 71

Hello, everybody. In this video, I'm gonna be showing you how to solve exercise 70 in chapter 13 Section four of calculus. Early transorbital. Now, in this problem, they want to verify whether or not the following statement is true with an explanation or false with the counter example in the statement that they were asking this for is whether or not the state the expression you crossed with U Cross V is equal to zero vector and this is false. If u and V are both non zero, which is the assumption given in the problem, then recall that by the definition of the cross product across product you ferocity, it's a vector that has a magnitude equal to the magnitude of you comes a bandit UV times a sine of the angle between U and V. So let's use this definition to find the magnitude of this left hand side. The magnitude of you cross U cross V is equal to the magnitude of you times a magnitude of U cross v times a sine of the angle between you and U cross V. But notice here that u cross V is always going to be orthogonal to you again by the definition of the cross product, and the sign of 90 degrees is one so we can forget about this term in this stuff. So now that we know that the magnitude of this vector equal to the magnitude of you, tens of magnitude of U cross V, we can further decompose the magnitude of this cross Britta into this depression that we have here and the result is magnitude of you squared. I'm so manitou to be times a sine of the angle between you be but both U and V or non zero. So these magnitudes or non zero as well end In most cases, the sign of the angle between these two vectors is not going to be zero. And this is only not true of their parallel. But that's not true in the general case. And so this magnitude is not necessarily equal to zero. And the only vector that has a magnitude of zero is zero. So we can conclude that this cross product you cross U cross V is not necessarily you hold it zero after. And that's how you sold exercise 70

Right A and B are two vectors and vector B is equal to the negative of vector A. Which of the following A Three D is trip A. The magnitude of B is equal to negative magnitude and a. B. A and B are perpendicular. See the angle or rather direction of vector A is equal to the angle or direction vector B plus 100 degrees or D after a prospector, B is equal to two times after A. So let's draw this out first to understand it. So we have two vectors of equal length A and B where the two vectors have opposite direction in this particular case, have shown that you could have been pointing at each other. So what are some things we notice about these vectors, first of all, since the magnitude of the vector, just length, it must be the magnitude of both A and B are equal. So not A B. They're parallel. They're not going to be a clear so yeah, not answer B. And we see the Director Edition A plus B. If we stack A and B on top of each other, but to be equal to the zero vector, they sum to zero because they're X components cancel each other. So it must be the only possible correct answer is C. The direction of A. In the direction of B plus 1 80.


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