Chapter 9: Q. 56 (page 631)

Find the area of the parallelogram with vertices $P_{1} = (1, 1, 1), P_{2} = (2, 3, 4), P_{3} = (6, 5, 2), a n d P_{4} = (7, 7, 5) .$

Short Answer

Expert verified

The area of the parallelogram is $18.2$ square units.

Step by step solution

Step 1. Given Information

The given vertices are $P_{1} = (1, 1, 1), P_{2} = (2, 3, 4), P_{3} = (6, 5, 2), a n d P_{4} = (7, 7, 5) .$

Step 2. Finding the area of the parallelogram

To find the area of the parallelogram we will use the formula $Area = ||u \times v|| .$

Let's find the two adjacent sides of a parallelogram

$u = \vec{P_{1} P_{2}} u = (2 - 1) i + (3 - 1) j + (4 - 1) k u = i + 2 j + 3 k v = \vec{P_{1} P_{3}} v = (6 - 1) i + (5 - 1) j + (2 - 1) k v = 5 i + 4 j + k$

Step 3. Solve

Let's find the cross product of the adjacent sides, the formula of the cross product is $v \times w = (b_{1} c_{2} - b_{2} c_{1}) i - (a_{1} c_{2} - a_{2} c_{1}) j + (a_{1} b_{2} - a_{2} b_{1}) k .$

So,

$u \times v = (i + 2 j + 3 k) \times (5 i + 4 j + k) u \times v = (2 \cdot 1 - 4 \cdot 3) i - (1 \cdot 1 - 5 \cdot 3) j + (1 \cdot 4 - 5 \cdot 2) k u \times v = - 10 i + 14 j - 6 k$

Now, use the formula of area of a parallelogram

$Area = ||u \times v|| Area = ||- 10 i + 14 j - 6 k|| Area = \sqrt{{(- 10)}^{2} + {(14)}^{2} + {(- 6)}^{2}} Area = \sqrt{332} Area \approx 18.2 square units$

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Find the area of the parallelogram with vertices $P_{1} = (1, 1, 1), P_{2} = (2, 3, 4), P_{3} = (6, 5, 2), a n d P_{4} = (7, 7, 5) .$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the area of the parallelogram

Step 3. Solve

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Find the area of the parallelogram with verticesP1=1,1,1,P2=2,3,4,P3=6,5,2,andP4=7,7,5.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the area of the parallelogram

Step 3. Solve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

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Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the area of the parallelogram with vertices $P_{1} = (1, 1, 1), P_{2} = (2, 3, 4), P_{3} = (6, 5, 2), a n d P_{4} = (7, 7, 5) .$