Chapter 9: Q. 45 (page 627)

In Problem, find the area of the parallelogram with one corner at P1 and adjacent sides $\vec{P 1 P 2} a n d \vec{P 1 P 3}$
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

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

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

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

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

Step3. Calculations of the area of a parallelogram

The area of parallelogram = $||u \times v||$

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

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

Short Answer

Step by step solution

Step1. Given information

step2. Calculations of the cross product

Step3. Calculations of the area of a parallelogram

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In Problem, find the area of the parallelogram with one corner at P1 and adjacent sidesP1P2→andP1P3→role="math" P1=0,0,0,P2=1,2,3,P3=-2,3,0

Short Answer

Step by step solution

Step1. Given information

step2. Calculations of the cross product

Step3. Calculations of the area of a parallelogram

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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