# Thread: parallelogram area with 3D vectors

1. ## parallelogram area with 3D vectors

How do I find the area of a parallelogram with the given vertices
K (1, 3, 2) L (1, 4, 4) M (4, 9, 4) N (4, 8, 2)

I have to use the cross product, but I'm pretty lost.

2. Originally Posted by acg716
How do I find the area of a parallelogram with the given vertices
K (1, 3, 2) L (1, 4, 4) M (4, 9, 4) N (4, 8, 2)

I have to use the cross product, but I'm pretty lost.
For any $\displaystyle \vec{u},\vec{v}\in\mathbb{R}^3$, $\displaystyle \|\vec{u}\times \vec{v}\|=\|\vec{u}\| \|\vec{v}\| |\sin(\vec{u},\vec{v})|$ equals the area of the parallelogram defined by the points $\displaystyle O,O+\vec{u},O+\vec{u}+\vec{v},O+\vec{v}$.

Thanks to the above, the area of $\displaystyle KLMN$ is $\displaystyle \|\overrightarrow{KL}\times\overrightarrow{KM}\|$: compute the vectors, the cross-product, and then the norm.

3. what do i do after i find the cross product of KL and KM ??

4. Originally Posted by acg716
what do i do after i find the cross product of KL and KM ??
Then (like I wrote earlier) the area is given by the norm of this vector: $\displaystyle \|(x,y,z)\|=\sqrt{x^2+y^2+z^2}$, where $\displaystyle (x,y,z)$ is to be replaced by the cross product.

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# area of a parallelogram from 3 vectors

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