# How do you find the area of the parallelogram with vertices k(1,2,3), l(1,3,6), m(3,8,6), and n(3,7,3)?

There are two ways, the first one ie VERY LONG and complicate, the second one VERY SHORT and easy, but we have to use the vectorial product.

The first one:

First of all, let's check if the shape is really a parallelogram:

A plane, perpendicular to a line, can be written:

So:

A line can be written as the system of three equation in parametric form:

or:

So:

Finally the area is:

The second one.

We can remember that the vectorial product between two vectors is a vector whose lenghts is the area of the parallelogram that has the two vector as two sides.

We can build the matrix:

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To find the area of the parallelogram formed by the given vertices, you can use the cross product of two vectors formed by the sides of the parallelogram. Let's label the vectors as (\overrightarrow{LM}) and (\overrightarrow{LN}). Then, the cross product of these vectors will give us a vector perpendicular to the parallelogram, and its magnitude will give us the area of the parallelogram.

First, calculate the vectors: (\overrightarrow{LM} = \overrightarrow{m} - \overrightarrow{l}) (\overrightarrow{LN} = \overrightarrow{n} - \overrightarrow{l})

Next, calculate the cross product of (\overrightarrow{LM}) and (\overrightarrow{LN}), denoted as (\overrightarrow{LM} \times \overrightarrow{LN}). This will give you a vector perpendicular to the parallelogram.

Finally, calculate the magnitude of (\overrightarrow{LM} \times \overrightarrow{LN}). This magnitude will give you the area of the parallelogram formed by the given vertices.

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