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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q9
11.4
![**Finding the Dot and Cross Product of Vectors**
**Problem Statement:**
Find \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \).
Given vectors:
\[
\mathbf{u} = \langle 10, 0, 0 \rangle
\]
\[
\mathbf{v} = \langle 1, 1, 1 \rangle
\]
\[
\mathbf{w} = \langle 0, 10, 10 \rangle
\]
**Solution Steps:**
1. **Calculate the Cross Product \( \mathbf{v} \times \mathbf{w} \)**:
The cross product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is given by:
\[
\mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle
\]
Applying to \( \mathbf{v} \) and \( \mathbf{w} \):
\[
\mathbf{v} \times \mathbf{w} = \langle 1 \cdot 10 - 1 \cdot 10, 1 \cdot 0 - 1 \cdot 10, 1 \cdot 10 - 1 \cdot 0 \rangle
\]
\[
= \langle 10 - 10, 0 - 10, 10 - 0 \rangle
= \langle 0, -10, 10 \rangle
\]
2. **Calculate the Dot Product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \)**:
The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd09d3726-e6d4-40fb-b56b-e32001dd8288%2F86d96e9a-2211-4f2d-a746-2088fb56134c%2F9lr69oj_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Dot and Cross Product of Vectors**
**Problem Statement:**
Find \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \).
Given vectors:
\[
\mathbf{u} = \langle 10, 0, 0 \rangle
\]
\[
\mathbf{v} = \langle 1, 1, 1 \rangle
\]
\[
\mathbf{w} = \langle 0, 10, 10 \rangle
\]
**Solution Steps:**
1. **Calculate the Cross Product \( \mathbf{v} \times \mathbf{w} \)**:
The cross product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is given by:
\[
\mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle
\]
Applying to \( \mathbf{v} \) and \( \mathbf{w} \):
\[
\mathbf{v} \times \mathbf{w} = \langle 1 \cdot 10 - 1 \cdot 10, 1 \cdot 0 - 1 \cdot 10, 1 \cdot 10 - 1 \cdot 0 \rangle
\]
\[
= \langle 10 - 10, 0 - 10, 10 - 0 \rangle
= \langle 0, -10, 10 \rangle
\]
2. **Calculate the Dot Product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \)**:
The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_
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