Chapter 9: Q. 50 (page 627)

In Problems 49–52, find the area of the parallelogram with vertices $P_{1}, P_{2}, P_{3}, P_{4}$
role="math" localid="1646760204418" $P_{1} = (2, 1, 1) P_{2} = (2, 3, 1) P_{3} = (- 2, 4, 1) P_{4} = (- 2, 6, 1)$

Short Answer

Expert verified

The area of the parallelogram is $8$ square units.

Step by step solution

Step 1. Given data

Vertices of the parallelogram are $P_{1} = (2, 1, 1) P_{2} = (2, 3, 1) P_{3} = (- 2, 4, 1) P_{4} = (- 2, 6, 1)$

The area of the Parallelogram is the magnitude of the cross product of the adjacent sides.

$A = ||u \times v||$

Step 2. Finding adjacent sides

Let's say adjacent sides are $u, v$ .

$u = \overset{⇀}{P_{1} P_{2}} \Rightarrow u = 0 i + 2 j + 0 k v = \overset{⇀}{P_{1} P_{3}} \Rightarrow v = - 4 i + 3 j + 0 k$

Step 3. Finding the area

The crossproduct:

$u \times v = |\begin{matrix} i & j & k \\ 0 & 2 & 0 \\ - 4 & 3 & 0 \end{matrix}| u \times v = |\begin{matrix} 2 & 0 \\ 3 & 0 \end{matrix}| i - |\begin{matrix} 0 & 0 \\ - 4 & 0 \end{matrix}| j + |\begin{matrix} 0 & 2 \\ - 4 & 3 \end{matrix}| k u \times v = 8 k$

Step 4. Finding the area

The area:

$A = ||u \times v|| A = ||8 k|| A = 8$

Therefore the area is $8$ square units.

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In Problems 49–52, find the area of the parallelogram with vertices $P_{1}, P_{2}, P_{3}, P_{4}$
role="math" localid="1646760204418" $P_{1} = (2, 1, 1) P_{2} = (2, 3, 1) P_{3} = (- 2, 4, 1) P_{4} = (- 2, 6, 1)$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Finding adjacent sides

Step 3. Finding the area

Step 4. Finding the area

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In Problems 49–52, find the area of the parallelogram with vertices P1,P2,P3,P4role="math" localid="1646760204418" P1=2,1,1P2=2,3,1P3=-2,4,1P4=-2,6,1

Short Answer

Step by step solution

Step 1. Given data

Step 2. Finding adjacent sides

Step 3. Finding the area

Step 4. Finding the area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 49–52, find the area of the parallelogram with vertices $P_{1}, P_{2}, P_{3}, P_{4}$
role="math" localid="1646760204418" $P_{1} = (2, 1, 1) P_{2} = (2, 3, 1) P_{3} = (- 2, 4, 1) P_{4} = (- 2, 6, 1)$