Question
Area of the parallelogram formed by the lines $y=m x, y=m x+1, y=$ $n x$ and $y=n x+1$ equals [2001S](a) $quad|m+n| /(m-n)^{2}$(b) $2 /|m+n|$(c) $1 /(|m+n|)$(d) $1 /(|m-n|)$
Area of the parallelogram formed by the lines $y=m x, y=m x+1, y=$ $n x$ and $y=n x+1$ equals [2001S] (a) $quad|m+n| /(m-n)^{2}$ (b) $2 /|m+n|$ (c) $1 /(|m+n|)$ (d) $1 /(|m-n|)$
Answers
Find the area of the parallelogram with vertices $ A (-3, 0), B (-1, 3), C (5, 2) $, and $ D (3, -1) $.
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In this problem, we will be working on finding the area of the parallelogram given the vectors, U equals to I minus J plus four k and the equals one half I plus two j minus three halves K. In the picture, I have Victor, you and Vector V and this is the parallelogram that they create. And we're trying to find the area of the shaded region. The area of a parallelogram determined by two vectors, is the magnitude of their cross product. So the first thing we have to find is the cross product. Have you, Doctor you and Vector V. So in review, we're going to use the Matrix to set that up, and our first row will be I J k. Our second row will be vector you and our third row will be vector V. And then we're going to set ourself up to take the determinant of the two by two vectors. So when I'm working on I, this is the two by two vector that's created. And then Jay, you see the two by two vector here and K there's too bad to Victor. Yeah, so now I'm going to find the determinant of each of these two by two vectors, So that'll be three halves minus eight. I negative three minus two J and four minus one half. Okay, which simplifies to negative 13 halves. I plus five J Plus seven halves K, which is the vector negative. 13 halves, five and seven halves. So our next step is to find the vector by Excuse me, find the magnitude of the cross product of those vectors, so we'll take the square root of negative. 13 have squared, plus five squared plus seven have squared, which is the square root of 1 59 over to. And then if we rationalize, we'll get the square root of 3 18 over to, which is approximately 8.92
Were given two vectors U and V, and our task is to find the area of the parallelogram formed by them. The way we will do so is by finding the magnitude of the cross product. So what I've done here is I have converted the I J K foreign vectors to their component form, So it's easier for us to see the different parts of the vectors. Okay, now, let's begin plugging stuff into the formula. So you two times V three. That's gonna be negative. One times negative one, which is just one minus you. Three times V two. That's just one times one one times one is just one and next part you three times if you want. You three is one B one is one That's just one You won times v three. You want his one? B three is negative ones. That's gonna be minus negative one. Now, for the last part of our cross product. You One times V two is just one times one giving us one minus you two times V one. That's a negative one. And now when we simplify this, we get zero two two No, what we have to do is find the magnitude of this vector. So well, we're gonna do is take the square root of the sum of the squares of these elements. So that's gonna be the area equals zero squared plus do squared close to squared. That's gonna be the square root of eight, which is equal to two times square root of two. So this is our final answer.
Who you calling about the area spent by the two vectors u N W you envy. So the area is equal to the you cost. They take in the absolute taking the norm Here on in this question were given the better you e code you 111 on the about to be equal to No, there was far. So if we charges, can the graph here you have in this form? So we have actually why on the G now the better new echoes you the 111 So 111 Uh huh. So it means that this could be the 111 So this could be the bit, uh, knew how and a better vehicle. Geezer's a phone. It will be this effectively here they found ballot program. It will be this form here. And that's what the area we're looking for. So the area you go, June the number on the V year gross fee. So if we go join the norm off, you cost when you can get equal with you. Ah, from the I get a good you know what, Farm Minister? We could your farm for the change. We can decode you form an is zero. It will be managed for for the Children and for the gonna coaches zero the phone get equal Thio square root off the foursquare plus false square Because you, the square the 32 I will go to the far square it with you.