Chapter 10: Q. 8 (page 823)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

What is meant by the parallelogram determined by vectors u and v in $ℝ^{3}$ ? 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 $ℝ^{3}$ 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 $||v|| \sin θ$ .

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

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

Short Answer

Step by step solution

Step 1. Given Information

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

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

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What is meant by the parallelogram determined by vectors u and v in ℝ3? How do you find the area of this parallelogram?

Short Answer

Step by step solution

Step 1. Given Information

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Discrete Mathematics

Pure Maths

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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