flightwingsd2

Answered

2022-06-26

I had a question about a problem that I was working on for my pre-calculus class.

Here's the problem statement:

The area of the parallelogram with vertices 0, $\u043c$, $w$, and $v+w$ is 34. Find the area of the parallelogram with vertices 0, $Av$, $Aw$, and $Av+Aw$, where

$A=\left(\begin{array}{cc}3& -5\\ -1& -3\end{array}\right).$

I got the answer by doing something very tedious. I set $v={\textstyle (}\genfrac{}{}{0ex}{}{17}{0}{\textstyle )}$ and $w={\textstyle (}\genfrac{}{}{0ex}{}{0}{2}{\textstyle )}$, and did some really crazy matrix multiplication and a lot of plotting points of GeoGebra to get the answer of: $\overline{){\displaystyle 476}}$.

Now, I'm 100% sure that was not the fastest way, can someone tell me the non-bash way to do the problem?

Here's the problem statement:

The area of the parallelogram with vertices 0, $\u043c$, $w$, and $v+w$ is 34. Find the area of the parallelogram with vertices 0, $Av$, $Aw$, and $Av+Aw$, where

$A=\left(\begin{array}{cc}3& -5\\ -1& -3\end{array}\right).$

I got the answer by doing something very tedious. I set $v={\textstyle (}\genfrac{}{}{0ex}{}{17}{0}{\textstyle )}$ and $w={\textstyle (}\genfrac{}{}{0ex}{}{0}{2}{\textstyle )}$, and did some really crazy matrix multiplication and a lot of plotting points of GeoGebra to get the answer of: $\overline{){\displaystyle 476}}$.

Now, I'm 100% sure that was not the fastest way, can someone tell me the non-bash way to do the problem?

Answer & Explanation

Angelo Murray

Expert

2022-06-27Added 23 answers

Hint: If $v=\left(\begin{array}{c}a\\ b\end{array}\right)$ and $w=\left(\begin{array}{c}c\\ d\end{array}\right)$ then the area of the original parallelogram is

$|det\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)|=|ad-bc|.$

Provide a similar expression for the area of the second parallelogram, and relate it to the above expression using properties of determinants.

$|det\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)|=|ad-bc|.$

Provide a similar expression for the area of the second parallelogram, and relate it to the above expression using properties of determinants.

sedeln5w

Expert

2022-06-28Added 7 answers

You're probably supposed to know that the absolute value of the determinant of $A$ is the area scale factor of the linear transformation represented by the matrix $A$. With this knowledge, you can easily solve the problem in your head.

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