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Chapter 2: Problem 7

(a) What is the sum in unit-vector notation of the two vectors \(\overrightarrow{\mathbf{a}}=5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{b}}=-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}} ?(b)\) What are the magni- tude and the direction of \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}\) ?

Short Answer

Expert verified
The sum of the two vectors in unit-vector notation is \(\overrightarrow{\mathbf{r}} = 2\hat{\mathbf{i}} +5\hat{\mathbf{j}}\). The magnitude of the resultant vector \(\overrightarrow{\mathbf{r}}\) is \(\sqrt{29}\), and its direction is approximately 68.20 degrees.

Step by step solution

01

Vector Addition in Unit-Vector Notation

Add the given vectors component by component. The i components are added separately from the j components. The sum for the i components is \(5 - 3 = 2\), and the sum for the j components is \(3 + 2 = 5\). So the resultant vector \(\overrightarrow{\mathbf{r}}= \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} = 2\hat{\mathbf{i}} + 5\hat{\mathbf{j}} \) .
02

Calculating the Magnitude of the Resultant Vector

The magnitude (length) of a vector in unit-vector notation is given by \(\sqrt{(i_{component})^{2} + (j_{component})^{2}}\). The magnitude of vector \(\overrightarrow{\mathbf{r}}\) is \(\sqrt{(2)^{2} + (5)^{2}} = \sqrt{29}\).
03

Finding the Direction of the Resultant Vector

The direction (angle) of a vector in unit-vector notation in the xy-plane is given by the formula \(\theta = \arctan(\frac{j_{component}}{i_{component}})\). The direction of vector \(\overrightarrow{\mathbf{r}}\) is \(\arctan(\frac{5}{2})\). Using a calculator, this comes out to approximately 68.20 degrees (in quadrants I and II).

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Most popular questions from this chapter

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