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Chapter 2: Problem 8

Two vectors are given by \(\overrightarrow{\mathbf{a}}=4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{b}}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}} .\) Find \((a) \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}},(b) \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}\), and \((c)\) a vector \(\overrightarrow{\mathbf{c}}\) such that \(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=0\)

Short Answer

Expert verified
The sum of vectors \( \overrightarrow{\mathbf{a}} \) and \( \overrightarrow{\mathbf{b}} \) is \( 3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 5\hat{\mathbf{k}} \). The difference of vectors \( \overrightarrow{\mathbf{a}} \) and \( \overrightarrow{\mathbf{b}} \) is \( 5\hat{\mathbf{i}}-4\hat{\mathbf{j}}-3\hat{\mathbf{k}} \). The vector \( \overrightarrow{\mathbf{c}} \) that satisfies the condition \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=0 \) is \( -5\hat{\mathbf{i}} + 4\hat{\mathbf{j}} + 3 \hat{\mathbf{k}} \).

Step by step solution

01

Adding Vectors

Vectors are added by component. So to find \( \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} \), the i, j, and k components of the two vectors should be added separately. \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} = (4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}) + (-\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}}) = 3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 5\hat{\mathbf{k}} \)
02

Subtracting Vectors

Like addition, vector subtraction is also done by subtracting corresponding components. So to find \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} \), subtract the i, j, and k components of vector b from vector a. \(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} = (4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}) - (-\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}}) = 5\hat{\mathbf{i}} - 4\hat{\mathbf{j}} - 3\hat{\mathbf{k}} \)
03

Find Vector \( \overrightarrow{\mathbf{c}} \)

To make the equation \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=0 \) correct, \( \overrightarrow{\mathbf{c}} \) should be equal to \( -\overrightarrow{\mathbf{a}} +\overrightarrow{\mathbf{b}} \).We find it by subtracting vector a from vector b: \( \overrightarrow{\mathbf{c}} = -\overrightarrow{\mathbf{a}} +\overrightarrow{\mathbf{b}} = -4\hat{\mathbf{i}} +3\hat{\mathbf{j}} -\hat{\mathbf{k}} + -\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}} = -5\hat{\mathbf{i}} + 4\hat{\mathbf{j}} + 3 \hat{\mathbf{k}} \).

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