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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 51 (page 627)

In Problems 49鈥�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

01

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.

$A = ||u 脳 v||$

02

Step 2. Finding adjacent sides

Let's say the adjacent sides are

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

03

Step 3. Finding the cross product

The cross product:

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

04

Step 4. Finding the area

The area is:

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

Therefore the area is $\sqrt{998}$ square units.

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Most popular questions from this chapter

If $\overset{鈬赌}{v} = 2 \overset{鈬赌}{i} - \overset{鈬赌}{j}$ and $\overset{鈬赌}{w} = x \overset{鈬赌}{i} + 3 \overset{鈬赌}{j}$ , find all numbers x for which $|\overset{鈬赌}{v} + \overset{鈬赌}{w}| = 5$ .

Identify and graph each polar equation. Verify your graph using a graphing utility.

$r = 3 + \cos 胃$

In Problems 61鈥�66, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.

$r = 3; r = 2 + 2 \cos 胃$

In Problems 39-54,the poplar coordinates of a point are given.Find rectangular coordinates of each point.

$(6, 150 掳)$

Write the expression in the standard form $a + i b$ .

${[3 (\cos 80 掳 + i \sin 80 掳)]}^{3}$

See all solutions

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