Chapter 9: Q. 48 (page 627)

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

Short Answer

Expert verified

The area of the parallelogram is : $\sqrt{217}$

Step by step solution

Step1.Giving information

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

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

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

Step2. calculation cross product of vectors

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

$u \times v = |\begin{matrix} i & j & k \\ 4 & 1 & - 3 \\ 4 & - 1 & 0 \end{matrix}| u \times v = [i |\begin{matrix} 1 & - 3 \\ - 1 & 0 \end{matrix}| - j |\begin{matrix} 4 & - 3 \\ 4 & 0 \end{matrix}| + k |\begin{matrix} 4 & 1 \\ 4 & - 1 \end{matrix}|] u \times v = - 3 i - 12 j - 8 k$

Step3. calculations of area

area of the parallelogram is : $||u \times v||$

$||u \times v|| = |- 3 i - 12 j - 8 k| ||u \times v|| = \sqrt{9 + 144 = 64} ||u \times v|| = \sqrt{217}$

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

Short Answer

Step by step solution

Step1.Giving information

Step2. calculation cross product of vectors

Step3. calculations of area

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

Short Answer

Step by step solution

Step1.Giving information

Step2. calculation cross product of vectors

Step3. calculations of area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Mechanics Maths

Logic and Functions

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

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