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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. 45 (page 627)

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

Short Answer

Expert verified

The area of the parallelogram is $\sqrt{166}$

Step by step solution

01

Step1. Given information

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

We know that the vector is formed by two points like

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

To given question

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

02

step2. Calculations of the 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

$u 脳 v = |\begin{matrix} i & j & k \\ 1 & 2 & 3 \\ - 2 & 3 & 0 \end{matrix}| u 脳 v = [i |\begin{matrix} 2 & 3 \\ 3 & 0 \end{matrix}| - j |\begin{matrix} 1 & 3 \\ - 2 & 0 \end{matrix}| + k |\begin{matrix} 1 & 2 \\ - 2 & 3 \end{matrix}|] u 脳 v = - 9 i - 6 j + 7 k$

03

Step3. Calculations of the area of a parallelogram

The area of parallelogram = $||u 脳 v||$

$||u 脳 v|| = |- 9 i - 6 j + 7 k| ||u 脳 v|| = \sqrt{81 + 36 + 49} ||u 脳 v|| = \sqrt{166}$

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Most popular questions from this chapter

In Problems $11 - 22$ , plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.

$- 2 + 3 i$ .

Match the graph with the polar equation.

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Write the expression in the standard form $a + b i$ .

${[\sqrt{3} (\cos \frac{5 蟺}{18} + i \sin \frac{5 蟺}{18})]}^{6}$

Identify and graph each polar equation. Verify your graph using a graphing utility.

$r = 3 \cos (4 胃)$

The standard equation of a circle with center at $(- 2, 5)$ and radius $3$ is ______ .

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