Let \(u = \left[ {\begin{array}{*{20}{c}}3\\0\end{array}} \right]\), and \(v = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Compute the area of the parallelogram

determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{array}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{array}} \right]\). How do they compare? Replace the first entry of v by an arbitrary number x, and repeat the problem. Draw a picture and explain what you find.

Obtain the vector by using the vectors and .

Thus, the vector is .

The graph of vectors , , , and 0 using arrows is shown below:

Here, the length of the base of the parallelogram is 3 units, and the height is 2 units.

The area of the parallelogram is calculated below:

Thus, the area is 6 square units.

Obtain the determinant of the vector .

Thus, .

The area computed by the graph and the determinant of vectors is the same.

It means the sides of the parallelogram adjacent to 0 defines the side of the parallelogram, and it is equal to the area of the parallelogram.

Replace 1 by x in the vector to make it .

Obtain the vector using the vectors and .

Replace 1 by x in the vector to make it .

Obtain the vector using the vectors and .

Here, the length of the base of the parallelogram is 3 units, and the height is 2 units.

The area of the parallelogram is calculated below:

Thus, the area is 6 square units. The obtained area is the same as in the first case because the base and height of the parallelogram are not changing.

So, it does not affect the area.

Obtain the determinant of vectors .

Thus, .