Chapter 9: Q. 49 (page 627)

In Problem, find the area of the parallelogram with vertices P1, P2, P3, and P4
$P 1 = (1, 1, 2), P 2 = (1, 2, 3), P 3 = (- 2, 3, 0), P 4 = (- 2, 4, 1)$

Short Answer

Expert verified

The area of the parallelogram is : $\sqrt{34}$

Step by step solution

Step1. Given information

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

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

$U = \vec{P 1 P 2} V = \vec{P 1 P 3} U = (1 - 1) i + (2 - 1) j + (3 - 2) k U = j + k l i k e t h a t V = - 3 i + 2 j - 2 k$

Step2. Calculating the orthogonal vector of u and v

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

$U = j + k V = - 3 i + 2 j - 2 k u \times v = |\begin{matrix} i & j & k \\ 0 & 1 & 1 \\ - 3 & 2 & - 2 \end{matrix}| u \times v = [i |\begin{matrix} 1 & 1 \\ 2 & - 2 \end{matrix}| - j |\begin{matrix} 0 & 1 \\ - 3 & - 2 \end{matrix}| + k |\begin{matrix} 0 & 1 \\ - 3 & 2 \end{matrix}|] u \times v = - 4 i - 3 j + 3 k$

Step3. Calculating the resultant of orthogonal vector

||u \times v|| = |- 4 i - 3 j + 3 k| ||u \times v|| = \sqrt{16 + 9 + 9} ||u \times v|| = \sqrt{34}

resultant of orthogonal vector = area of given parallelogram

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problem, find the area of the parallelogram with vertices P1, P2, P3, and P4
$P 1 = (1, 1, 2), P 2 = (1, 2, 3), P 3 = (- 2, 3, 0), P 4 = (- 2, 4, 1)$

Short Answer

Step by step solution

Step1. Given information

Step2. Calculating the orthogonal vector of u and v

Step3. Calculating the resultant of orthogonal vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Logic and Functions

Pure Maths

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problem, find the area of the parallelogram with vertices P1, P2, P3, and P4P1=1,1,2,P2=1,2,3,P3=-2,3,0,P4=-2,4,1

Short Answer

Step by step solution

Step1. Given information

Step2. Calculating the orthogonal vector of u and v

Step3. Calculating the resultant of orthogonal vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Logic and Functions

Pure Maths

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problem, find the area of the parallelogram with vertices P1, P2, P3, and P4
$P 1 = (1, 1, 2), P 2 = (1, 2, 3), P 3 = (- 2, 3, 0), P 4 = (- 2, 4, 1)$