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A brother and sister are playing in the woods, when suddenly the brother realizes that they are separated. The last place he remembers seeing his sister is at a particularly large tree. The brother traveled d1=22.0 m at θ1=22.0∘ from the tree then turned and traveled d2=10.0 m at θ2=131∘. Meanwhile, the sister traveled d3=19.0 m at an angle of θ3=−113∘,from the tree. The angles are given with respect to east with counterclockwise being defined as positive

**Figure Explanation:** The diagram shows an \( xy \)-plane with vectors representing the position of a boy relative to a tree. The tree is located at the origin \((0,0)\). Two vectors are shown: one along the positive \( x \)-axis and another pointing in the direction away from the tree towards the boy. The exact angle and lengths of the vectors are not specified. **Part A - Position of the Boy** *Question:* Using the parallelogram law to add these vectors geometrically, determine the displacement of the boy relative to the tree. [(Figure 2)](URL_to_Figure_2) Express your answers, separated by a comma, to three significant figures. Enter the angle measured counterclockwise from the positive \( x \)-axis. --- *Explanation for Students:* To solve this problem, you will apply the parallelogram law. This involves combining the given vectors by constructing a parallelogram where the vectors are adjacent sides, then finding the diagonal of the parallelogram, which represents the resultant vector. Steps to follow: 1. **Draw the Vectors:** Start by plotting the given vectors from the tree (origin). 2. **Construct the Parallelogram:** Depending on the direction and magnitude of the vectors, draw a parallelogram. 3. **Find the Diagonal:** The diagonal's length and direction will give you the displacement vector of the boy relative to the tree. 4. **Calculate and Express Result:** Present your final answer in terms of magnitude and direction (angle counterclockwise from the positive \( x \)-axis), ensuring that you round your answer to three significant figures.
Transcribed Image Text:**Figure Explanation:** The diagram shows an \( xy \)-plane with vectors representing the position of a boy relative to a tree. The tree is located at the origin \((0,0)\). Two vectors are shown: one along the positive \( x \)-axis and another pointing in the direction away from the tree towards the boy. The exact angle and lengths of the vectors are not specified. **Part A - Position of the Boy** *Question:* Using the parallelogram law to add these vectors geometrically, determine the displacement of the boy relative to the tree. [(Figure 2)](URL_to_Figure_2) Express your answers, separated by a comma, to three significant figures. Enter the angle measured counterclockwise from the positive \( x \)-axis. --- *Explanation for Students:* To solve this problem, you will apply the parallelogram law. This involves combining the given vectors by constructing a parallelogram where the vectors are adjacent sides, then finding the diagonal of the parallelogram, which represents the resultant vector. Steps to follow: 1. **Draw the Vectors:** Start by plotting the given vectors from the tree (origin). 2. **Construct the Parallelogram:** Depending on the direction and magnitude of the vectors, draw a parallelogram. 3. **Find the Diagonal:** The diagonal's length and direction will give you the displacement vector of the boy relative to the tree. 4. **Calculate and Express Result:** Present your final answer in terms of magnitude and direction (angle counterclockwise from the positive \( x \)-axis), ensuring that you round your answer to three significant figures.
In the provided image, a coordinate axis system (x and y axes) is depicted with three vectors, \( \mathbf{d_1} \), \( \mathbf{d_2} \), and \( \mathbf{d_3} \). These vectors represent directions in a plane. The vectors are displayed with arrows indicating their orientation and magnitude. ### Detailed Explanation 1. **Coordinate Axes:** - The horizontal axis is labeled as \( x \). - The vertical axis is labeled as \( y \). 2. **Vectors:** - **\( \mathbf{d_1} \)**: - This vector originates from an initial point and points upwards and to the right. - It forms an angle \( \theta_1 \) with the vector \( \mathbf{d_2} \). - **\( \mathbf{d_2} \)**: - This vector starts from the same initial point as \( \mathbf{d_1} \). - It is directed upwards and slightly to the right. - It forms an angle \( \theta_2 \) with the positive \( x \)-axis. - **\( \mathbf{d_3} \)**: - This vector points downwards and slightly to the left. - It forms an angle \( \theta_3 \) with the vector \( \mathbf{d_1} \). 3. **Angles:** - **\( \theta_1 \)**: The angle between vectors \( \mathbf{d_1} \) and \( \mathbf{d_2} \). - **\( \theta_2 \)**: The angle between vector \( \mathbf{d_2} \) and the positive \( x \)-axis. - **\( \theta_3 \)**: The angle between vectors \( \mathbf{d_1} \) and \( \mathbf{d_3} \). This diagram is crucial for understanding vector directions and the angles between them in a two-dimensional plane. Understanding these relationships is fundamental in fields such as physics and engineering, where vector analysis is commonly employed.
Transcribed Image Text:In the provided image, a coordinate axis system (x and y axes) is depicted with three vectors, \( \mathbf{d_1} \), \( \mathbf{d_2} \), and \( \mathbf{d_3} \). These vectors represent directions in a plane. The vectors are displayed with arrows indicating their orientation and magnitude. ### Detailed Explanation 1. **Coordinate Axes:** - The horizontal axis is labeled as \( x \). - The vertical axis is labeled as \( y \). 2. **Vectors:** - **\( \mathbf{d_1} \)**: - This vector originates from an initial point and points upwards and to the right. - It forms an angle \( \theta_1 \) with the vector \( \mathbf{d_2} \). - **\( \mathbf{d_2} \)**: - This vector starts from the same initial point as \( \mathbf{d_1} \). - It is directed upwards and slightly to the right. - It forms an angle \( \theta_2 \) with the positive \( x \)-axis. - **\( \mathbf{d_3} \)**: - This vector points downwards and slightly to the left. - It forms an angle \( \theta_3 \) with the vector \( \mathbf{d_1} \). 3. **Angles:** - **\( \theta_1 \)**: The angle between vectors \( \mathbf{d_1} \) and \( \mathbf{d_2} \). - **\( \theta_2 \)**: The angle between vector \( \mathbf{d_2} \) and the positive \( x \)-axis. - **\( \theta_3 \)**: The angle between vectors \( \mathbf{d_1} \) and \( \mathbf{d_3} \). This diagram is crucial for understanding vector directions and the angles between them in a two-dimensional plane. Understanding these relationships is fundamental in fields such as physics and engineering, where vector analysis is commonly employed.
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