Find the area of the triangle with vertices P (0, 0, -3), Q (4, 2,0), R(3, 3, 1)V86140 9O V862.

Answered: Find the area of the triangle with…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.1: Rectangular Coordinate Systems

Problem 15E

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Question

Find the area of the triangle with vertices P (0, 0, -3), Q (4, 2,0), R(3, 3, 1)
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Expert Solution

Step 1

Given points are $P (0, 0, - 3), Q (4, 2, 0), R (3, 3, 1)$ .

Obtain the vectors as below.

$\begin{array}{rcl} \overset{⇀}{P Q} & = & (4 - 0) i + (2 - 0) j + (0 + 3) k \\ = & 4 i + 2 j + 3 k \\ \overset{⇀}{P R} & = & (3 - 0) i + (3 - 0) j + (1 + 3) k \\ = & 3 i + 3 j + 4 k \end{array}$

Step 2

Obtain the cross product.

$\begin{array}{rcl} \overset{⇀}{P Q} \times \overset{⇀}{P R} & = & |\begin{matrix} i & j & k \\ 4 & 2 & 3 \\ 3 & 3 & 4 \end{matrix}| \\ = & i (8 - 9) - j (16 - 9) + k (12 - 6) \\ = & i + 25 j + 6 k \\ |\overset{⇀}{P Q} \times \overset{⇀}{P R}| & = & \sqrt{1^{2} + 25^{2} + 6^{2}} \\ = & \sqrt{1 + 625 + 36} \\ = & \sqrt{662} \end{array}$