Question

In Problems 49-52, find the area of the parallelogram with vertices $P_1, P_2, P_3$, and $P_4$. $\begin{aligned} P_1 & =(2,1,1), \quad P_2=(2,3,1), \quad P_3=(-2,4,1), \\ P_4 & =(-2,6,1)\end{aligned}$

Name: Problem 50, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2021-08-03T16:35:06-08:00
Duration: 3 min 32 s
Channel: Babita Kumari
Description: In Problems 49-52, find the area of the parallelogram with vertices P1, P2, P3, and P4. P1 =(2,1,1), P2=(2,3,1), P3=(-2,4,1), P4 =(-2,6,1)

   In Problems 49-52, find the area of the parallelogram with vertices $P_1, P_2, P_3$, and $P_4$.
$\begin{aligned} P_1 & =(2,1,1), \quad P_2=(2,3,1), \quad P_3=(-2,4,1), \\ P_4 & =(-2,6,1)\end{aligned}$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 8, Problem 50

Instant Answer

Solved by Expert Babita Kumari

08/03/2021

Step 1

The vertices of the parallelogram are given as $ P_1 = (2,1,1) $, $ P_2 = (2,3,1) $, $ P_3 = (-2,4,1) $, and $ P_4 = (-2,6,1) $. Show more…

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In Problems 49-52, find the area of the parallelogram with vertices $P_1, P_2, P_3$, and $P_4$. $\begin{aligned} P_1 & =(2,1,1), \quad P_2=(2,3,1), \quad P_3=(-2,4,1), \\ P_4 & =(-2,6,1)\end{aligned}$

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Key Concepts

Vector Subtraction

Vector subtraction is used to determine the edge vectors of the parallelogram. By subtracting the coordinates of one vertex from those of an adjacent vertex, you obtain a vector that represents one side of the parallelogram in the coordinate space.

Cross Product

The cross product of two vectors in three-dimensional space yields a new vector that is perpendicular to the plane containing the original vectors. Importantly, the magnitude of this cross product is equal to the area of the parallelogram spanned by the two vectors.

Magnitude of a Vector

The magnitude of a vector, computed as the square root of the sum of the squares of its components, provides the length of the vector. In the context of the cross product, the magnitude represents the actual area of the parallelogram defined by the two initial edge vectors.

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