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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 47 (page 627)

In Problem, find the area of the parallelogram with one corner at P1 and adjacent sides
$\overset{鈫�}{P 1 P 2} a n d \overset{鈫�}{P 1 P 3}$
$P 1 = (1, 2, 0), P 2 = (- 2, 3, 4), P 3 = (0, - 2, 3)$

Short Answer

Expert verified

The area of the parallelogram is : $\sqrt{555}$

Step by step solution

01

Step1. Giving information

three points $P 1 (1, 2, 0), P 2 (- 2, 3, 4), P 3 (0, - 2, 3)$

We know that the vector is formed by two points like

A = ({x_{2} - x}_{1}) i + (y_{2} - y_{1}) j + (z_{2} - z_{1}) k

To given question

$U = \overset{鈫�}{P 1 P 2} U = (- 2 - 1) i + (3 - 2) j + (4 - 0) k U = - 3 i + j + 4 k l i k e t h a t v = - i - 4 j + 3 k$

02

Step2. Calculation of cross product

We know that to find the area of a parallelogram by two vectors. it's necessary to find an orthogonal vector, for that

$u 脳 v = |\begin{matrix} i & j & k \\ - 3 & 1 & 4 \\ - 1 & - 4 & 3 \end{matrix}| u 脳 v = [i |\begin{matrix} 1 & 4 \\ - 4 & 3 \end{matrix}| + j |\begin{matrix} - 3 & 4 \\ - 1 & 3 \end{matrix}| + k |\begin{matrix} - 3 & 1 \\ - 1 & - 4 \end{matrix}|] u 脳 v = 19 i + 5 j + 13 k$

03

Step3. Area of parallelogram

$||u 脳 v|| = |19 i + 5 j + 13 k| ||u 脳 v|| = \sqrt{361 + 25 + 169} ||u 脳 v|| = \sqrt{555}$

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Most popular questions from this chapter

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