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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![**Problem Statement:**
Let \(\vec{a} = 5\vec{j} + \vec{k}\) and \(\vec{b} = -2\vec{i} + \vec{j} + 2\vec{k}\). Find \(-3\vec{a} - 3\vec{b}\).
**Step-by-Step Solution:**
1. **Define the vectors:**
\[
\vec{a} = 5\vec{j} + \vec{k}
\]
\[
\vec{b} = -2\vec{i} + \vec{j} + 2\vec{k}
\]
2. **Multiply \(\vec{a}\) by \(-3\):**
\[
-3\vec{a} = -3(5\vec{j} + \vec{k}) = -15\vec{j} - 3\vec{k}
\]
3. **Multiply \(\vec{b}\) by \(-3\):**
\[
-3\vec{b} = -3(-2\vec{i} + \vec{j} + 2\vec{k}) = 6\vec{i} - 3\vec{j} - 6\vec{k}
\]
4. **Add \(-3\vec{a}\) and \(-3\vec{b}\):**
\[
-3\vec{a} - 3\vec{b} = (-15\vec{j} - 3\vec{k}) + (6\vec{i} - 3\vec{j} - 6\vec{k})
\]
Combine like terms:
\[
= 6\vec{i} + (-15\vec{j} - 3\vec{j}) + (-3\vec{k} - 6\vec{k})
\]
\[
= 6\vec{i} - 18\vec{j} - 9\vec{k}
\]
So, the resultant vector is:
\[
-3\vec{a} - 3\vec{b} = 6\vec{i} - 18\vec{j} - 9\vec{k}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55d3466e-6a31-4aff-972b-e33941523058%2F51f1d279-58b2-4fb1-ac76-3ec5987f933d%2F2edwx0j_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \(\vec{a} = 5\vec{j} + \vec{k}\) and \(\vec{b} = -2\vec{i} + \vec{j} + 2\vec{k}\). Find \(-3\vec{a} - 3\vec{b}\).
**Step-by-Step Solution:**
1. **Define the vectors:**
\[
\vec{a} = 5\vec{j} + \vec{k}
\]
\[
\vec{b} = -2\vec{i} + \vec{j} + 2\vec{k}
\]
2. **Multiply \(\vec{a}\) by \(-3\):**
\[
-3\vec{a} = -3(5\vec{j} + \vec{k}) = -15\vec{j} - 3\vec{k}
\]
3. **Multiply \(\vec{b}\) by \(-3\):**
\[
-3\vec{b} = -3(-2\vec{i} + \vec{j} + 2\vec{k}) = 6\vec{i} - 3\vec{j} - 6\vec{k}
\]
4. **Add \(-3\vec{a}\) and \(-3\vec{b}\):**
\[
-3\vec{a} - 3\vec{b} = (-15\vec{j} - 3\vec{k}) + (6\vec{i} - 3\vec{j} - 6\vec{k})
\]
Combine like terms:
\[
= 6\vec{i} + (-15\vec{j} - 3\vec{j}) + (-3\vec{k} - 6\vec{k})
\]
\[
= 6\vec{i} - 18\vec{j} - 9\vec{k}
\]
So, the resultant vector is:
\[
-3\vec{a} - 3\vec{b} = 6\vec{i} - 18\vec{j} - 9\vec{k}
\]
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