Q. 8 - Calculus

Question

What is meant by the parallelogram determined by vectors u and v in ${\mathrm{ℝ}}^3$? How do you find the area of this parallelogram?

Step-by-Step Solution

Answer

The area of the parallelogram is $||u||||v||\sin \theta =||u\times v||$.

Step 1. Given Information

What is meant by the parallelogram determined by vectors u and v in ${\mathrm{ℝ}}^3$? How do you find the area of this parallelogram?

Step 2. We first consider the case where u and v are parallel vectors.

Here the parallelogram determined by u and v is degenerate; that is, it is collapsed into a line segment and has zero area. 

By Theorem the cross product of two parallel vectors u and v in ${\mathrm{ℝ}}^3$ is $u\times v=0$ so $||u\times v||=||0||=0$

If u and v are not parallel, then the area of the parallelogram determined by u and v is the product of the length of one of the sides and the distance between that side and the opposite side.

Step 3. In our parallelogram, u is the length of one side.

In the parallelogram from the figure, we see that the distance from that side to the opposite side is $||v||\sin \theta$.

Therefore the area of the parallelogram is $||u||||v||\sin \theta =||u\times v||$.