Chapter 10.3, Problem 4E

To determine

To fill in the blank

Answer to Problem 4E

The area of a parallelogram with $u\text{ and }v$ as adjacent sides is given by $\underset{\_}{\parallel u\times v\parallel }$ .

Explanation of Solution

Given information : 

  $u\text{ and }v$ are adjacent sides of parallelogram.

Concept used:

Concept of cross product

Explanation: 

Area of parallelogram $=base\times height$

Height of parallelogram is perpendicular to the base of parallelogram,

Given $uand\text{ v}$ are adjacent sides of parallelogram

Magnitude of adjacent sides is given by $\parallel u\parallel and\parallel v\parallel$

Illustrating given information with help of diagram given below

  

  $\begin{array}{l}\sin \theta =\frac{perpendicular}{hypotenuse}=\frac{height}{\parallel v\parallel }\\ \Rightarrow height=\parallel v\parallel \times \sin \theta \\ Therefore,\\ \text{Area of parallelogram=}\parallel u\parallel \cdot \parallel v\parallel \sin \theta =\parallel u\times v\parallel \\ \left(because\parallel a\times b\parallel =\parallel a\parallel .\parallel b\parallel \sin \theta \right)\end{array}$

Hence,

The area of a parallelogram with $u\text{ and }v$ as adjacent sides is given by $\underset{\_}{\parallel u\times v\parallel }$ .