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So I've been working on problem 7 in 12.4. Following the book on page 315 about parallelograms.

"Given two vectors, we can put them tail to tail and form a parallelogram...The height of the parallelogram, h, is |A|$ \sin(\theta)$ , and the base is |B|, so the area of the parallelogram is |A||B|$ \sin(\theta)$, exactly the magnitude of |A x B|."

The problem states "7. Find the area of the parallelogram with vertices (0, 0), (1, 2), (3, 7), and (2, 5)."

When I sketched out the parallelogram with the given points, it looked like the base of the parallelogram would be the vector from the origin to (1,2) (B), and the top vector would be (2,5)(A). (I'm not sure if this is where my mistake was, or if it was in trying to find $\theta$.)

Using those two points as the vectors, I ended up with |A| = $\sqrt{29}$ and |B| =$ \sqrt{5}. $

I then used the dot product |A| * |B| = |A||B|$\cos$($\theta$) to attempt to find theta. Plugging everything in I ended up with $$ \theta = \cos^{-1}(\frac {12} {\sqrt{29} \sqrt{5}}) $$

Plugging that into the area equation from the book, I ended up with..

$$ area = \sqrt{29}\sqrt{5}(\sin(\cos^{-1}(\frac {12} {\sqrt{29}\sqrt{5}})))$$

Which unless I did wrong in my calculator gave me 0. But the back of the book showed that the answer should have been 1. So I'm not exactly sure where I went wrong, or if I did the whole problem wrong. So if anybody could give some input that would be fantastic!

Thanks!

So I've been working on problem 7 in 12.4. Following the book on page 315 about parallelograms.

"Given two vectors, we can put them tail to tail and form a parallelogram...The height of the parallelogram, h, is |A|$ \sin(\theta)$ , and the base is |B|, so the area of the parallelogram is |A||B|$ \sin(\theta)$, exactly the magnitude of |A x B|."

The problem states "7. Find the area of the parallelogram with vertices (0, 0), (1, 2), (3, 7), and (2, 5)."

When I sketched out the parallelogram with the given points, it looked like the base of the parallelogram would be the vector from the origin to (1,2) (B), and the top vector would be (2,5)(A). (I'm not sure if this is where my mistake was, or if it was in trying to find $\theta$.)

Using those two points as the vectors, I ended up with |A| = $\sqrt{29}$ and |B| =$ \sqrt{5}. $

I then used the dot product |A| * |B| = |A||B|$\cos$($\theta$) to attempt to find theta. Plugging everything in I ended up with $$ \theta = \cos^{-1}(\frac {12} {\sqrt{29} \sqrt{5}}) $$

Plugging that into the area equation from the book, I ended up with..

$$ area = \sqrt{29}\sqrt{5}(\sin(\cos^{-1}(\frac {12} {\sqrt{29}\sqrt{5}})))$$

Which unless I did wrong in my calculator gave me 0. But the back of the book showed that the answer should have been 1. So I'm not exactly sure where I went wrong, or if I did the whole problem wrong. So if anybody could give some input that would be fantastic!

Thanks!

So I've been working on problem 7 in 12.4. Following the book on page 315 about parallelograms.

"Given two vectors, we can put them tail to tail and form a parallelogram...The height of the parallelogram, h, is |A|$ \sin(\theta)$ , and the base is |B|, so the area of the parallelogram is |A||B|$ \sin(\theta)$, exactly the magnitude of |A x B|."

The problem states "7. Find the area of the parallelogram with vertices (0, 0), (1, 2), (3, 7), and (2, 5)."

When I sketched out the parallelogram with the given points, it looked like the base of the parallelogram would be the vector from the origin to (1,2) (B), and the top vector would be (2,5)(A). (I'm not sure if this is where my mistake was, or if it was in trying to find $\theta$.)

Using those two points as the vectors, I ended up with |A| = $\sqrt{29}$ and |B| =$ \sqrt{5}. $

I then used the dot product |A| * |B| = |A||B|$\cos$($\theta$) to attempt to find theta. Plugging everything in I ended up with $$ \theta = \cos^{-1}(\frac {12} {\sqrt{29} \sqrt{5}}) $$

Plugging that into the area equation from the book, I ended up with..

$$ area = \sqrt{29}\sqrt{5}(\sin(\cos^{-1}(\frac {12} {\sqrt{29}\sqrt{5}})))$$

Which unless I did wrong in my calculator gave me 0. But the back of the book showed that the answer should have been 1. So I'm not exactly sure where I went wrong, or if I did the whole problem wrong. So if anybody could give some input that would be fantastic!

Thanks!