Chapter 9: Q. 51 (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="1646761740205" $P_{1} = (1, 2, - 1) P_{2} = (4, 2, - 3) P_{3} = (6, - 5, 2) P_{4} = (9, - 5, 0)$

Short Answer

Expert verified

The area of the parallelogram is $\sqrt{998}$ square units.

Step by step solution

Step 1. Given data

Vertices of the parallelogram are

$P_{1} = (1, 2, - 1) P_{2} = (4, 2, - 3) P_{3} = (6, - 5, 2) P_{4} = (9, - 5, 0)$

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 the adjacent sides are

$u = \overset{⇀}{P_{1} P_{2}} \Rightarrow u = 3 i + 0 j - 2 k v = \overset{⇀}{P_{1} P_{3}} \Rightarrow v = 5 i - 7 j + 3 k$

Step 3. Finding the cross product

The cross product:

$u \times v = |\begin{matrix} i & j & k \\ 3 & 0 & - 2 \\ 5 & - 7 & 3 \end{matrix}| u \times v = |\begin{matrix} 0 & - 2 \\ - 7 & 3 \end{matrix}| i - |\begin{matrix} 3 & - 2 \\ 5 & 3 \end{matrix}| j + |\begin{matrix} 3 & 0 \\ 5 & - 7 \end{matrix}| k u \times v = - 14 i - 19 j - 21 k$

Step 4. Finding the area

The area is:

$A = ||u \times v|| A = ||- 14 i - 19 - 21 k|| A = |\sqrt{{(- 14)}^{2} + {(- 19)}^{2} + {(- 21)}^{2}}| A = \sqrt{998}$

Therefore the area is $\sqrt{998}$ 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="1646761740205" $P_{1} = (1, 2, - 1) P_{2} = (4, 2, - 3) P_{3} = (6, - 5, 2) P_{4} = (9, - 5, 0)$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Finding adjacent sides

Step 3. Finding the cross product

Step 4. Finding the area

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In Problems 49–52, find the area of the parallelogram with verticesP1,P2,P3,P4role="math" localid="1646761740205" P1=1,2,-1P2=4,2,-3P3=6,-5,2P4=9,-5,0

Short Answer

Step by step solution

Step 1. Given data

Step 2. Finding adjacent sides

Step 3. Finding the cross product

Step 4. Finding the area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Probability and Statistics

Geometry

Statistics

Applied Mathematics

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="1646761740205" $P_{1} = (1, 2, - 1) P_{2} = (4, 2, - 3) P_{3} = (6, - 5, 2) P_{4} = (9, - 5, 0)$