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a =i +j +2k,b =i+k,č = j +k.

a =i +j +2k,b =i+k,č = j +k.

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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find the volume of the parallelepiped
with edges ⃗a , ⃗b , ⃗c .

**19.** \[\mathbf{a} = \mathbf{i} + \mathbf{j} + 2\mathbf{k}, \; \mathbf{b} = \mathbf{i} + \mathbf{k}, \; \mathbf{c} = \mathbf{j} + \mathbf{k}.\] **Explanation:** This mathematical expression defines three vectors in three-dimensional space, using unit vectors: - **\(\mathbf{a}\)** is represented as \(\mathbf{i} + \mathbf{j} + 2\mathbf{k}\), which means it has components of 1 in the x-direction, 1 in the y-direction, and 2 in the z-direction. - **\(\mathbf{b}\)** is represented as \(\mathbf{i} + \mathbf{k}\), indicating it has components of 1 in the x-direction and 1 in the z-direction. - **\(\mathbf{c}\)** is represented as \(\mathbf{j} + \mathbf{k}\), meaning it has components of 1 in the y-direction and 1 in the z-direction. These vectors can be visualized in a Cartesian coordinate system where \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the unit vectors along the x, y, and z axes respectively. This representation is useful for studying vector operations such as addition, subtraction, and dot or cross products in physics and engineering.
Transcribed Image Text:**19.** \[\mathbf{a} = \mathbf{i} + \mathbf{j} + 2\mathbf{k}, \; \mathbf{b} = \mathbf{i} + \mathbf{k}, \; \mathbf{c} = \mathbf{j} + \mathbf{k}.\] **Explanation:** This mathematical expression defines three vectors in three-dimensional space, using unit vectors: - **\(\mathbf{a}\)** is represented as \(\mathbf{i} + \mathbf{j} + 2\mathbf{k}\), which means it has components of 1 in the x-direction, 1 in the y-direction, and 2 in the z-direction. - **\(\mathbf{b}\)** is represented as \(\mathbf{i} + \mathbf{k}\), indicating it has components of 1 in the x-direction and 1 in the z-direction. - **\(\mathbf{c}\)** is represented as \(\mathbf{j} + \mathbf{k}\), meaning it has components of 1 in the y-direction and 1 in the z-direction. These vectors can be visualized in a Cartesian coordinate system where \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the unit vectors along the x, y, and z axes respectively. This representation is useful for studying vector operations such as addition, subtraction, and dot or cross products in physics and engineering.
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