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4. Find an equation for the plane through (1, 3, 4), (2, 2, 3), and (3, 2, 1).

4. Find an equation for the plane through (1, 3, 4), (2, 2, 3), and (3, 2, 1).

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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### Vector and Plane Exercises 1. **For the vectors \(\mathbf{v} = \langle 1, -1, 2 \rangle\) and \(\mathbf{u} = \langle 3, 1, -1 \rangle\):** a. Find \(\mathbf{v} \cdot \mathbf{u}\) b. Find \(\|\mathbf{v}\|\) and \(\|\mathbf{u}\|\) c. Find the angle between \(\mathbf{v}\) and \(\mathbf{u}\). 2. **Determine whether the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel, orthogonal, or neither for the given \(\mathbf{a}\) and \(\mathbf{b}\):** a. \(\mathbf{a} = \langle 1, 2, 3 \rangle\) and \(\mathbf{b} = \langle 5, -1, -1 \rangle\) b. \(\mathbf{a} = \langle 1, 2, 3 \rangle\) and \(\mathbf{b} = \langle 4, 8, 12 \rangle\) c. \(\mathbf{a} = \langle 1, 2, 3 \rangle\) and \(\mathbf{b} = \langle 2, 4, 3 \rangle\) 3. **Find the cross product \(\mathbf{u} \times \mathbf{v}\) when \(\mathbf{u} = \langle 0, 1, 1 \rangle\) and \(\mathbf{v} = \langle 1, 0, 1 \rangle\).** 4. **Find an equation for the plane through the points \((1, 3, 4)\), \((2, 2, 3)\), and \((3, 2, 1)\).** 5. **Find parametric equations for the line through the points \((1, 3, 0)\) and \((2, 4, 4)\).**
Transcribed Image Text:### Vector and Plane Exercises 1. **For the vectors \(\mathbf{v} = \langle 1, -1, 2 \rangle\) and \(\mathbf{u} = \langle 3, 1, -1 \rangle\):** a. Find \(\mathbf{v} \cdot \mathbf{u}\) b. Find \(\|\mathbf{v}\|\) and \(\|\mathbf{u}\|\) c. Find the angle between \(\mathbf{v}\) and \(\mathbf{u}\). 2. **Determine whether the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel, orthogonal, or neither for the given \(\mathbf{a}\) and \(\mathbf{b}\):** a. \(\mathbf{a} = \langle 1, 2, 3 \rangle\) and \(\mathbf{b} = \langle 5, -1, -1 \rangle\) b. \(\mathbf{a} = \langle 1, 2, 3 \rangle\) and \(\mathbf{b} = \langle 4, 8, 12 \rangle\) c. \(\mathbf{a} = \langle 1, 2, 3 \rangle\) and \(\mathbf{b} = \langle 2, 4, 3 \rangle\) 3. **Find the cross product \(\mathbf{u} \times \mathbf{v}\) when \(\mathbf{u} = \langle 0, 1, 1 \rangle\) and \(\mathbf{v} = \langle 1, 0, 1 \rangle\).** 4. **Find an equation for the plane through the points \((1, 3, 4)\), \((2, 2, 3)\), and \((3, 2, 1)\).** 5. **Find parametric equations for the line through the points \((1, 3, 0)\) and \((2, 4, 4)\).**
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