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Find the area of the parallelogram with vertices P C0, 0, 0) QC 31-374) R(3₁-1₁ ~51 5C6₁ -4,-97

Find the area of the parallelogram with vertices P C0, 0, 0) QC 31-374) R(3₁-1₁ ~51 5C6₁ -4,-97

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Title: Calculating the Area of a Parallelogram with Given Vertices** **Instructions:** To find the area of a parallelogram with the following vertices in a 3-dimensional space: - Vertex P: \( (0, 0, 0) \) - Vertex Q: \( (3, -3, -4) \) - Vertex R: \( (3, -1, -5) \) - Vertex S: \( (6, -4, -9) \) **Steps:** 1. **Represent the Position Vectors:** - Vector \( \mathbf{PQ} \): From P to Q. - Vector \( \mathbf{PR} \): From P to R. 2. **Calculate Vectors:** - \( \mathbf{PQ} = \langle 3-0, -3-0, -4-0 \rangle = \langle 3, -3, -4 \rangle \) - \( \mathbf{PR} = \langle 3-0, -1-0, -5-0 \rangle = \langle 3, -1, -5 \rangle \) 3. **Cross Product:** - Find the cross product \( \mathbf{PQ} \times \mathbf{PR} \). 4. **Area Calculation:** - The magnitude of \( \mathbf{PQ} \times \mathbf{PR} \) gives the area of the parallelogram. **Diagram Explained:** There is a basic sketch of a parallelogram on the page, aimed at representing a generic quadrilateral shape illustrating the concept of vertices and vectors within a 3D space.
Transcribed Image Text:**Title: Calculating the Area of a Parallelogram with Given Vertices** **Instructions:** To find the area of a parallelogram with the following vertices in a 3-dimensional space: - Vertex P: \( (0, 0, 0) \) - Vertex Q: \( (3, -3, -4) \) - Vertex R: \( (3, -1, -5) \) - Vertex S: \( (6, -4, -9) \) **Steps:** 1. **Represent the Position Vectors:** - Vector \( \mathbf{PQ} \): From P to Q. - Vector \( \mathbf{PR} \): From P to R. 2. **Calculate Vectors:** - \( \mathbf{PQ} = \langle 3-0, -3-0, -4-0 \rangle = \langle 3, -3, -4 \rangle \) - \( \mathbf{PR} = \langle 3-0, -1-0, -5-0 \rangle = \langle 3, -1, -5 \rangle \) 3. **Cross Product:** - Find the cross product \( \mathbf{PQ} \times \mathbf{PR} \). 4. **Area Calculation:** - The magnitude of \( \mathbf{PQ} \times \mathbf{PR} \) gives the area of the parallelogram. **Diagram Explained:** There is a basic sketch of a parallelogram on the page, aimed at representing a generic quadrilateral shape illustrating the concept of vertices and vectors within a 3D space.
**Find the Area of the Parallelogram with Vertices** Vertices: - \( P(0,0,0) \) - \( Q(3,-3,-4) \) - \( R(3,-1,-5) \) - \( S(6,-4,-9) \) **Diagram:** The image contains a roughly sketched rectangle representing the parallelogram. However, in a 3D coordinate space, the vertices define the shape as a parallelogram, not necessarily a rectangle.
Transcribed Image Text:**Find the Area of the Parallelogram with Vertices** Vertices: - \( P(0,0,0) \) - \( Q(3,-3,-4) \) - \( R(3,-1,-5) \) - \( S(6,-4,-9) \) **Diagram:** The image contains a roughly sketched rectangle representing the parallelogram. However, in a 3D coordinate space, the vertices define the shape as a parallelogram, not necessarily a rectangle.
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The area of the parallelogram ABCD is 

Area=AB→×AC→

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