Question 1078312
.
<U>Theorem</U>


<pre>
    If  <B>u</B> = {{{(matrix(2,1, a, c))}}}&nbsp; and &nbsp;<B>v</B> = {{{(matrix(2,1, b, d))}}}&nbsp; are vectors in a coordinate plane, then the area of the parallelogram which is built on these vectors 

    as on sides is equal to the modulus of the determinant, &nbsp;|det {{{(matrix(2,2, a, b, c, d))}}}|, &nbsp;of the &nbsp;2x2-matrix &nbsp;<B>A</B> = {{{(matrix(2,2, a, b, c, d))}}}&nbsp; whose columns are the given vectors.
</pre>

For the proof of this theorem see the lesson 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=>Determinant of a 2x2-matrix and the area of a parallelogram and a triangle</A> 

in this site.


<pre>
Your vector U is the vertical side of the parallelogram from the point (-1,-5) to the point (-1,1).

It has component form U = (-1-(-1),1-(-5)) = (-1+1, 1+6) = (0,7).


Your vector V is the sloped side of the parallelogram from the point (-1,-5) to the point (4,5).

It has component form V = (4-(-1),5-(-5)) = (4+1, 5+5) = (5,10).


Now make a matrix A whose columns are the components of the vectors U and V:


    A = {{{(matrix(2,2, 0,5, 7,10))}}}.


Then take its determinant  det(A) = det {{{(matrix(2,2, 0,5, 7,10))}}} = -5*7 = -35.


Finally, take the <U>modulus</U> of the determinant, i.e. its absolute value.

You will get the area of your parallelogram


    S = | det (A) | = |-35| = 35.


<U>Answer</U>.  The area of the parallelogram is 35 square units.
</pre>

 *** SOLVED ***



There are lessons in this site relevant to this theme:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/What-is-a-matrix.lesson>What is a matrix?</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-2x2-matrix.lesson>Determinant of a 2x2-matrix</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-2x2-matrix_and_the_area_of_a_parallelogram_and_a_triangle.lesson>Determinant of a 2x2-matrix and the area of a parallelogram and a triangle</A>.  



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
&nbsp;&nbsp;&nbsp;&nbsp; "<U>2x2-Matrices, determinants, Cramer's rule for systems in two unknowns</U>"