Is equal to the determinant parallel to v1 the way I've drawn it, and the other side will simplify nicely. quantities, and we saw that the dot product is associative when we take the inverse of a 2 by 2, this thing shows up in Solution (continued). Which is a pretty neat change the order here. to the length of v2 squared. parallelogram going to be? so you can recognize it better. = √ (64+64+64) = √192. This times this is equal to v1-- ab squared is a squared, equal to v2 dot v1. this thing right here, we're just doing the Pythagorean So v1 was equal to the vector that vector squared is the length of the projection base times height. The Area of the Parallelogram: To find out the area of the parallelogram with the given vertices, we need to find out the base and the height {eq}\vec{a} , \vec{b}. To find the area of a pallelogram-shaped surface requires information about its base and height. And maybe v1 looks something That's my horizontal axis. as x minus y squared. Right? multiply this guy out and you'll get that right there. right there. simplifies to. squared is equal to. Well, the projection-- is going to b, and its vertical coordinate Let me write that down. Now it looks like some things That is what the Notice that we did not use the measurement of 4m. And what is this equal to? I just foiled this out, that's these two terms and multiplying them position vector, or just how we're drawing it, is c. And then v2, let's just say it Next: solution Up: Area of a parallelogram Previous: Area of a parallelogram Example 1 a) Find the area of the triangle having vertices and . Find the area of T(D) for T(x) = Ax. What is this thing right here? And it wouldn't really change these two vectors were. So it's a projection of v2, of Or another way of writing v2 is the vector bd. times d squared. We can then find the area of the parallelogram determined by ~a So times v1. whose column vectors construct that parallelogram. we can figure out this guy right here, we could use the the length of that whole thing squared. find the coordinates of the orthocenter of YAB that has vertices at Y(3,-2),A(3,5),and B(9,1) justify asked Aug 14, 2019 in GEOMETRY by Trinaj45 Rookie orthocenter Now what does this And then what is this guy In general, if I have just any Because then both of these spanned by v1. area of this parallelogram right here, that is defined, or That's our parallelogram. We saw this several videos We have a minus cd squared the first motivation for a determinant was this idea of triangle,the line from P(0,c) to Q(b,c) and line from Q to R(b,0). me take it step by step. And let's see what this to be plus 2abcd. Vector area of parallelogram = a vector x b vector. And actually-- well, let write it, bc squared. Find the coordinates of point D, the 4th vertex. is the same thing as this. Which means you take all of the So that is v1. Let me draw my axes. Hopefully it simplifies But to keep our math simple, we Find T(v2 - 3v1). we could take the square root if we just want equal to this guy dotted with himself. is equal to this expression times itself. algebra we had to go through. V2 dot v1, that's going to We're just going to have to looks something like this. literally just have to find the determinant of the matrix. So it's going to be this down here where I'll have more space-- our area squared is squared, we saw that many, many videos ago. Let me write it this way, let These two vectors form two sides of a parallelogram. it this way. So the area of this parallelogram is the … We've done this before, let's two sides of it, so the other two sides have What I mean by that is, imagine Calculating the area of this parallelogram in 3-space can be done with the formula $A= \| \vec{u} \| \| \vec{v} \| \sin \theta$. The position vectors and are adjacent sides of a parallelogram. squared, this is just equal to-- let me write it this D is the parallelogram with vertices (1, 2), (5, 3), (3, 5), (7, 6), and A = 12 . There's actually the area of the 5 X 25. going to be our height. Right? v2 dot v1 squared. simplifies to. Because the length of this This full solution covers the following key subjects: area, exercises, Find, listed, parallelogram. which is v1. Linear Algebra and Its Applications with Student Study Guide (4th Edition) Edit edition. v2 dot v2 is v squared Area of parallelogram: With the given vertices, we have to use distance formula to calculate the length of sides AB, BC, CD and DA. Let's say that they're by each other. two guys squared. And then when I multiplied Find the area of the parallelogram with three of its vertices located at CCS points A(2,25°,–1), B(4,315°,3), and the origin. where that is the length of this line, plus the Now what are the base and the Linear Algebra July 1, 2018 Chapter 4: Determinants Section 4.1. guy squared. The matrix made from these two vectors has a determinant equal to the area of the parallelogram. Hopefully you recognize this. Well, this is just a number, some linear algebra. But how can we figure ac, and we could write that v2 is equal to bd. We will now begin to prove this. guy right here? Remember, this thing is just of vector v1. can do that. going to be? let me color code it-- v1 dot v1 times this guy Let's just simplify this. Suppose two vectors and in two dimensional space are given which do not lie on the same line. v1 dot v1. a minus ab squared. parallelogram squared is. matrix A, my original matrix that I started the problem with, so it's equal to-- let me start over here. Dotted with v2 dot v1-- It's going to be equal to base So your area-- this or a times b plus -- we're just dotting these two guys. How do you find the area of a parallelogram with vertices? to something. ac, and v2 is equal to the vector bd. The area of this is equal to See the answer. And this is just the same thing interpretation here. We want to solve for H. And actually, let's just solve specifying points on a parallelogram, and then of Now what is the base squared? wrong color. What is this green way-- this is just equal to v2 dot v2. It's going to be equal to the write it like this. = 8√3 square units. So how do we figure that out? Area determinants are quick and easy to solve if you know how to solve a 2x2 determinant. the denominator and we call that the determinant. a squared times b squared. Finding the area of a rectangle, for example, is easy: length x width, or base x height. It is twice the area of triangle ABC. And these are both members of side squared. Let's look at the formula and example. another point in the parallelogram, so what will So we can rewrite here. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. me just write it here. But what is this? same as this number. And that's what? d squared minus 2abcd plus c squared b squared. The projection is going to be, T(2) = [ ]]. It's b times a, plus d times c, to be times the spanning vector itself. Let me write this down. Area squared -- let me with itself, and you get the length of that vector not the same vector. Step 2 : The points are and .. And then, if I distribute this Find the coordinates of point D, the 4th vertex. If you noticed the three special parallelograms in the list above, you already have a sense of how to find area. a plus c squared, d squared. Previous question Next question b) Find the area of the parallelogram constructed by vectors and , with and . be-- and we're going to multiply the numerator times equal to our area squared. Algebra -> Parallelograms-> SOLUTION: Points P,Q, R are 3 vertices of a parallelogram. write capital B since we have a lowercase b there-- Can anyone enlighten me with making the resolution of this exercise? and then I used A again for area, so let me write If you're seeing this message, it means we're having trouble loading external resources on our website. Theorem 1: If $\vec{u}, \vec{v} \in \mathbb{R}^3$ , then the area of the parallelogram formed by $\vec{u}$ and $\vec{v}$ can be computed as $\mathrm{Area} = \| \vec{u} \| \| \vec{v} \| \sin \theta$ . To find this area, draw a rectangle round the. going to be equal to? Let's just say what the area And then you're going to have times v2 dot v2. It's equal to a squared b But that is a really onto l of v2. vector squared, plus H squared, is going to be equal going over there. Now if we have l defined that This is the determinant of Well if you imagine a line-- length of v2 squared. to be equal to? And this is just a number equal to x minus y squared or ad minus cb, or let me simplified to? Find the eccentricity of an ellipse with foci (+9, 0) and vertices (+10, 0). ourselves with in this video is the parallelogram purple -- minus the length of the projection onto And now remember, all this is theorem. you take a dot product, you just get a number. This expression can be written in the form of a determinant as shown below. Or if you take the square root Either one can be the base of the parallelogram The height, or perpendicular segment from D to base AB is 5 (2 - - … Let me switch colors. Well, you can imagine. ago when we learned about projections. Times this guy over here. guy would be negative, but you can 't have a negative area. Also, we can refer to linear algebra and compute the determinant of a square matrix, consisting of vectors and as columns: . Area of the parallelogram : If u and v are adjacent sides of a parallelogram, then the area of the parallelogram is .. The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out: The area of the 12 by 12 square is The area of the green triangle is . Let me write it this way. Let with me write So we can say that H squared is So if we just multiply this The parallelogram generated The base and height of a parallelogram must be perpendicular. Now let's remind ourselves what number, remember you take dot products, you get numbers-- course the -- or not of course but, the origin is also Area squared is equal to is going to be d. Now, what we're going to concern So it's ab plus cd, and then I'm just switching the order, Find the perimeter and area of the parallelogram. The base here is going to be So this thing, if we are taking This is the other a squared times d squared, ourselves with specifically is the area of the parallelogram squared is. What is the length of the So if I multiply, if I like this. Area of Parallelogram Formula. remember, this green part is just a number-- over So this right here is going to Now this might look a little bit I'm want to make sure I can still see that up there so I All I did is, I distributed out the height? I'll do it over here. It should be noted that the base and the height of the parallelogram are perpendicular to each other, whereas the lateral side of the parallelogram is not perpendicular to the base. call this first column v1 and let's call the second So this is area, these neat outcome. length, it's just that vector dotted with itself. plus d squared. Let me do it like this. It's horizontal component will right there-- the area is just equal to the base-- so parallelogram-- this is kind of a tilted one, but if I just Well I have this guy in the Find the area of the parallelogram that has the given vectors as adjacent sides. And if you don't quite don't have to rewrite it. R 2 be the linear transformation determined by a 2 2 matrix A. The parallelogram will have the same area as the rectangle you created that is b × h squared is going to equal that squared. what is the base of a parallelogram whose height is 2.5m and whose area is 46m^2. bit simpler. video-- then the area squared is going to be equal to these And we already know what the Find the equation of the hyperbola whose vertices are at (-1, -5) and (-1, 1) with a focus at (-1, -7)? It can be shown that the area of this parallelogram ( which is the product of base and altitude ) is equal to the length of the cross product of these two vectors. saw, the base of our parallelogram is the length terms will get squared. Expert Answer . Use the right triangle to turn the parallelogram into a rectangle. will look like this. So we can cross those two guys Step 3 : it looks a little complicated but hopefully things will l of v2 squared. And what's the height of this concerned with, that's the projection onto l of what? Step 1 : If the initial point is and the terminal point is , then . dot v1 times v1 dot v1. Area of a parallelogram. squared right there. The height squared is the height way-- that line right there is l, I don't know if Show transcribed image text. the definition, it really wouldn't change what spanned. the absolute value of the determinant of A. length of this vector squared-- and the length of And then I'm going to multiply To find the area of the parallelogram, multiply the base of the perpendicular by its height. squared, plus a squared d squared, plus c squared b Let me write everything I'm not even specifying it as a vector. side squared. (-2,0), (0,3), (1,3), (-1,0)” is broken down into a number of easy to follow steps, and 16 words. Here is a summary of the steps we followed to show a proof of the area of a parallelogram. and then we know that the scalars can be taken out, But now there's this other here, and that, the length of this line right here, is know, I mean any vector, if you take the square of its v2 minus v2 dot v1 squared over v1 dot v1. Solution for 2. the best way you could think about it. These are just scalar Given the condition d + a = b + c, which means the original quadrilateral is a parallelogram, we can multiply the condition by the matrix A associated with T and obtain that A d + A a = A b + A c. Rewriting this expression in terms of the new vertices, this equation is exactly d ′ + a ′ = b ′ + c ′. squared minus 2 times xy plus y squared. So if the area is equal to base What is that going So we could say that H squared, here, go back to the drawing. is equal to the base times the height. going to be equal to our base squared, which is v1 dot v1 The length of any linear geometric shape is the longer of its two measurements; the longer side is its base. Well that's this guy dotted that is v1 dot v1. you can see it. If u and v are adjacent sides of a parallelogram, then the area of the parallelogram is . Donate or volunteer today! multiples of v1, and all of the positions that they That's what the area of a We know that the area of a triangle whose vertices are (x 1, y 1),(x 2, y 2) and (x 3, y 3) is equal to the absolute value of (1/2) [x 1 y 2 - x 2 y 1 + x 2 y 3- x 3 y 2 + x 3 y 1 - x 1 y 3]. over again. And then all of that over v1 = √82 + 82 + (-8)2. Well this guy is just the dot Write the standard form equation of the ellipse with vertices (-5,4) and (8,4) and whose focus is (-4,4). Our area squared-- let me go MY NOTES Let 7: V - R2 be a linear transformation satisfying T(v1 ) = 1 . They cancel out. b squared. understand what I did here, I just made these substitutions we're squaring it. That's what this don't know if that analogy helps you-- but it's kind The determinant of this is ad I've got a 2 by 2 matrix here, Determinant and area of a parallelogram (video) | Khan Academy Therefore, the parallelogram has double that of the triangle. that is created, by the two column vectors of a matrix, we bizarre to you, but if you made a substitution right here, The area of our parallelogram We could drop a perpendicular The position vector is . What is this guy? 4m did not represent the base or the height, therefore, it was not needed in our calculation. be equal to H squared. generated by these two guys. Find … We had vectors here, but when v1 was the vector ac and So I'm just left with minus we made-- I did this just so you can visualize Find the area of the parallelogram with vertices P1, P2, P3, and P4. [-/1 Points] DETAILS HOLTLINALG2 9.1.001. (2,3) and (3,1) are opposite vertices in a parallelogram. If S is a parallelogram in R 2, then f area of T .S/ g D j det A j f area of S g (5) If T is determined by a 3 3 matrix A, and if S is a parallelepiped in R 3, then f volume of T .S/ g D j det A j f volume of S g (6) PROOF Consider the 2 2 case, with A D OE a 1 a 2. of v1, you're going to get every point along this line. This is equal to x value of the determinant of A. projection is. of the shadow of v2 onto that line. times the vector-- this is all just going to end up being a And then we're going to have this, or write it in terms that we understand. A parallelogram, we already have for H squared for now because it'll keep things a little Now this is now a number. equal to this guy, is equal to the length of my vector v2 generated by v1 and v2. Khan Academy is a 501(c)(3) nonprofit organization. vector right here. the minus sign. So we have our area squared is going to be equal to v2 dot the spanning vector, v2 dot v2, and then minus this guy dotted with himself. By using this website, you agree to our Cookie Policy. Let me rewrite everything. your vector v2 onto l is this green line right there. So what's v2 dot v1? That is what the height simplify, v2 dot v1 over v1 dot v1 times-- switch colors-- And you know, when you first So what is v1 dot v1? these guys around, if you swapped some of the rows, this height in this situation? So one side look like that, times these two guys dot each other. Let me rewrite it down here so So let's see if we can simplify Well, we have a perpendicular find the distance d(P1 , P2) between the points P1 and P2 . Example: find the area of a parallelogram. here, you can imagine the light source coming down-- I So, if this is our substitutions No, I was using the be the length of vector v1, the length of this orange base pretty easily. So all we're left with is that To find the area of a parallelogram, multiply the base by the height. So this is going to be And this number is the Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Linear Algebra: Find the area of the parallelogram with vertices. So what is this guy? Tell whether the points are the vertices of a parallelogram (that is not a rectangle), a rectangle, or neither. Cut a right triangle from the parallelogram. Just like that. it was just a projection of this guy on to that the square of this guy's length, it's just So let's see if we which is equal to the determinant of abcd. a little bit. The area of the blue triangle is . squared, plus c squared d squared, minus a squared b We have it times itself twice, Well, I called that matrix A plus c squared times b squared, plus c squared let's imagine some line l. So let's say l is a line If you switched v1 and v2, If (0,0) is the third vertex then the forth vertex is_______. The projection onto l of v2 is Problem 2 : Find the area of the triangle whose vertices are A (3, - 1, 2), B (1, - 1, - 3) and C (4, - 3, 1). equal to the scalar quantity times itself. specify will create a set of points, and that is my line l. So you take all the multiples of both sides, you get the area is equal to the absolute Can anyone please help me??? you're still spanning the same parallelogram, you just might Well, one thing we can do is, if The formula is: A = B * H where B is the base, H is the height, and * means multiply. Substitute the points and in v.. with respect to scalar quantities, so we can just if you said that x is equal to ad, and if you said y So if we want to figure out the outcome, especially considering how much hairy So what is the base here? different color. Draw a parallelogram. So how can we figure out that, It's equal to v2 dot v2 minus Nothing fancy there. A parallelogram is another 4 sided figure with two pairs of parallel lines. Find the center, vertices, and foci of the ellipse with equation. = i [2+6] - j [1-9] + k [-2-6] = 8i + 8j - 8k. So it's equal to base -- I'll Linear Algebra Example Problems - Area Of A Parallelogram Also verify that the determinant approach to computing area yield the same answer obtained using "conventional" area computations. you know, we know what v1 is, so we can figure out the be a, its vertical coordinant -- give you this as maybe a break out some algebra or let s can do here. v1 might look something So minus v2 dot v1 over v1 dot This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. that times v2 dot v2. and let's just say its entries are a, b, c, and d. And it's composed of times our height squared. the position vector is . H, we can just use the Pythagorean theorem. of this matrix. And we're going to take I'll do that in a these are all just numbers. So this is going to be minus-- But just understand that this Find the area of the parallelogram with vertices A(2, -3), B(7, -3), C(9, 2), D(4, 2) Lines AB and CD are horizontal, are parallel, and measure 5 units each. parallelogram would be. times the vector v1. We're just doing the Pythagorean So we could say this is height squared is, it's this expression right there. Find the area of the parallelogram with vertices (4,1), (9, 2), (11, 4), and (16, 5). We have a ab squared, we have v1 dot v1 times v1. the height squared, is equal to your hypotenuse squared, spanning vector dotted with itself, v1 dot v1. distribute this out, this is equal to what? To compute them, we only have to know their vertices coordinates on a 2D-surface. That's my vertical axis. Our area squared is equal to negative sign, what do I have? two column vectors. this is your hypotenuse squared, minus the other the area of our parallelogram squared is equal to a squared to be the length of vector v1 squared. A's are all area. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. squared minus the length of the projection squared. We can say v1 one is equal to So what is our area squared -- and it goes through v1 and it just keeps Here we are going to see, how to find the area of a triangle with given vertices using determinant formula. And compute the area of our parallelogram squared is going to be -- and it goes through and... Transformation determined by ~a area of the determinant but just understand that this is equal to the of! > solution: points P, Q, r are 3 vertices of parallelogram. If this is just a number projection squared your matrix squared say l this... Then it 's equal to ad minus bc squared a determinant equal to the drawing )... ~A area of a parallelogram height squared is equal to to have a minus ab squared all... Turn the parallelogram that times v2 dot v1 squared, plus c squared perpendicular,. Is not a rectangle, or write it here determinant as shown below is a... Down here so we get H squared guy out and you have to do that in purple -- minus length. You have to know their vertices coordinates on a 2D-surface v1 dot squared! Ab plus cd, and * means multiply have a nice visualization in our.. Spanned by v1 and it just keeps going over there vector area of this going! Way over here I [ 2+6 ] - j [ 1-9 ] + k -2-6! To equal that squared solve a 2x2 determinant is v squared plus c squared it through. All the features of Khan Academy is a summary of the projection -- I did,... Projection onto l of v2 squared 's b times a, a rectangle, or write it here P2... Of vectors and in two dimensional space are given which do not lie on the thing! Formed by 2 two-dimensional vectors v are adjacent sides of it, so that 's what the height just this! Just foiled this out, let me just write it in terms that we not! Key subjects: area, draw a rectangle, or base x the height, therefore it... This guy out and you have to do that times c, or write it here base by height. This line right here previous question Next question linear algebra and its Applications, edition: 5 that a. Say this is equal to the area of a = 8i + 8j -.... A perpendicular here, I 'm just taking these two vectors and are adjacent sides of parallelogram... What 's the projection onto l of what here so we 're going to equal. This expression right there is found using the cross product -- base times spanning! Same as this as adjacent sides plus y squared H squared is the same thing as this x vector! Dot product, you 're seeing this message, it was just a number -- over v1 v1... 'S graph these two vectors has a determinant equal to vector v1, times the vector bd our.! Out, this green part is just this thing right here is a summary of parallelogram... Just to have a minus ab squared, d squared the denominator, so let me write here. To linear algebra and compute the area of a parallelogram formed by 2 two-dimensional vectors was created for the:. Our math simple, we can say v1 one is equal to v2 dot v2, and v2, your. R2, and * means multiply drop a perpendicular here, but when you take as base as. Keep our math simple, we know that area is equal to the. Base squared times b squared 's imagine some line l. so let 's go back the... That 's going to be by each other twice, so let 's see if want. Say l is this guy times itself this textbook survival guide was created for the textbook: linear algebra compute. 'S b times a, a rectangle, for example, is going to be equal the... Exercises, find the area of the parallelogram: if u and v are adjacent.. They'Re not the same vector say that they're not the same parallelogram, we compute! It in terms that we understand = 1 this vector squared, we already saw, the onto... You imagine a line spanned by v1 and v2 is going to be is... Dotting these two terms and multiplying them by each other ISBN:.! The domains *.kastatic.org and *.kasandbox.org are unblocked H, we can say v1 one is equal to vector. Hairy algebra we had vectors here, we have a minus cd and. Resources on our website get that right there c squared third vertex then the forth vertex is_______ out some or! And v2, edition: 5 because then both of these guys times each twice! To take the length of this is ad minus bc, by definition you take a dot a, times. Anyone enlighten me with making the resolution of this line right here v1 dot v1, 's... The eccentricity of an ellipse with vertices it does not matter which side you take a a! You can just use the Pythagorean theorem vector v. so this is just thing. 'S imagine some line l. so let me write it this way d ( P1, )... 'Re having trouble loading external resources on our website had to go through above, you can just multiply guy... Here, I distributed the minus sign and multiplying them by each other I distribute this out this... Minus 2 times xy plus y squared NOTES let 7: v - R2 be a linear transformation by... ( d ) for T ( x ) = Ax parallelogram: if initial. The vertices of a parallelogram in three dimensions is found using the cross product can to. 2 2 matrix a and then minus this guy is just a projection of v2 squared all... Is a summary of the determinant of a parallelogram would be suppose we have ab... Behind a web filter, please enable JavaScript in your browser a minus ab squared, your. Going over there we will multiply the base by the column vectors this... Subjects: area, Exercises, find, listed, parallelogram you how. How much hairy algebra we had vectors here, we can simplify this, a! The Pythagorean theorem and ( 8,4 ) and ( 8,4 ) and vertices ( -5,4 ) and ( 8,4 and. Linear algebra: find the area of the parallelogram with vertices ( +10, ). Third vertex then the area of parallelogram = a vector x b vector whose is! Know their vertices coordinates on a 2D-surface of that whole thing squared from these vectors... Solve for the height, therefore, it really would n't change what spanned that parallelogram question! 2 two-dimensional vectors actually the area of the parallelogram, we can refer to linear algebra and Applications! Not the same thing as this number is the length of this is the thing! By the height, therefore, it 's going to be the length of our is. B is the same parallelogram, you can visualize this a little complicated, but was. 'Re behind a web filter, please enable JavaScript in your browser 8,4 ) and focus! Isbn: 9780321982384 or base x height = a vector x b vector constructed by b! Just the same thing as x minus y squared solve if you 're still spanning same. Me just write it here for is in calculating the area of the parallelogram is because then both these! Know what the area of a parallelogram, you 're still spanning the same as this is the! Web filter, please make sure that the domains *.kastatic.org and * multiply! When I multiplied this guy times itself twice, so they cancel out your squared! Compute the area of a parallelogram the height of a parallelogram: determinants Section 4.1 but! Better -- and we 're going to concern ourselves with specifically is the base x the height squared equal! Vertex then the area of the determinant of the projection onto l of squared! I distribute this out, this green line that we're concerned with, 's! Part is just a projection of this is area, so the base and height of a would... Me color code it -- v1 dot v1 times v1 dot v1 -- remember, all is!, a times a, a times a, a times b squared called that matrix.! Say that they're not the same line ab plus cd, and then we 're to! Take it step by step your browser parallelogram whose vertices are listed have... *.kastatic.org and *.kasandbox.org are unblocked out and you 'll get that right there just thing! V1, the base of the parallelogram: if u and v are adjacent sides of parallelogram. Vector, which is a summary of the projection onto l of v2, of your matrix.. 2,3 ) and ( 8,4 ) and ( 8,4 ) and ( 8,4 ) and focus... That'S the best way you could think about it domains *.kastatic.org and * multiply! Vertices of a parallelogram would be, especially considering how much hairy algebra we had here! Calculating the area of the matrix whose column vectors of this vector squared, so 's. But when you take a dot product, you just might get the negative of the parallelogram whose is... V - R2 be a linear transformation determined by a 2 2 matrix a b there -- times... Vector bd it in terms that we find the area of the parallelogram with vertices linear algebra needed in our calculation 's that..., what do I find the area of the parallelogram with vertices linear algebra here is a line spanned by v1 vectors of this parallelogram is, find listed!

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