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Chapter 2: Problem 3

Vector \(\overrightarrow{\mathbf{a}}\) has a magnitude of \(5.2\) units and is directed east. Vector \(\overrightarrow{\mathbf{b}}\) has a magnitude of \(4.3\) units and is directed \(35^{\circ}\) west of north. By constructing vector diagrams, find the magnitudes and directions of (a) \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}\), and \((\mathrm{b}) \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}\).

Short Answer

Expert verified
The magnitudes and directions of \( \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} \) and \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} \) can be obtained by converting the vectors from polar to Cartesian form, performing vector addition and subtraction, and converting the results back to polar form.

Step by step solution

01

Convert Polar to Cartesian Form

For a vector directed east and west, this is similar to positive and negative x-direction respectively. And a vector directed north equals positive y-direction. So, convert polar vectors \( \overrightarrow{\mathbf{a}} \) and \( \overrightarrow{\mathbf{b}} \) to Cartesian form. \( \overrightarrow{\mathbf{a}} = 5.2\hat{i} \) (as it is directed towards the east) and \( \overrightarrow{\mathbf{b}} = 4.3cos(35^{\circ})(-\hat{i}) + 4.3sin(35^{\circ})\hat{j} \) (as it is directed 35 degrees west of north).
02

Perform Vector Addition and Subtraction

Calculate \( \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} \) and \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} \) using the Cartesian vectors obtained from the previous step.
03

Convert Resultant Vectors Back to Polar Form

Convert \( \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} \) and \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} \) back to their polar forms to find the magnitude and direction of the resultant vectors. The magnitude is given by the formula \( \sqrt{x^{2}+y^{2}} \) and the direction by \( \text{arctan}(y/x) \), where x and y are the components of the resultant vectors in the Cartesian plane.

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