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Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Magnitude Angle ||M| = e = 150° %3D

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Magnitude Angle ||M| = e = 150° %3D

Calculus: Early Transcendentals
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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**Find the component form of v given its magnitude and the angle it makes with the positive x-axis.** **Magnitude:** \[ \|v\| = \frac{9}{7} \] **Angle:** \[ \theta = 150^\circ \] \[ \text{Sketch } \mathbf{v}: \] (Insert the sketch of vector \(\mathbf{v}\) here) --- To find the component form of vector \(\mathbf{v}\), we use the magnitude \(\|v\|\) and the angle \(\theta\) it makes with the positive x-axis. The components of \(\mathbf{v}\) can be calculated using the following formulas: \[ v_x = \|v\| \cos(\theta) \] \[ v_y = \|v\| \sin(\theta) \] Given: \[ \|v\| = \frac{9}{7} \] \[ \theta = 150^\circ \] We calculate the components as follows: **For the x-component:** \[ v_x = \frac{9}{7} \cos(150^\circ) \] \[ \cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} \] \[ v_x = \frac{9}{7} \cdot -\frac{\sqrt{3}}{2} = -\frac{9\sqrt{3}}{14} \] **For the y-component:** \[ v_y = \frac{9}{7} \sin(150^\circ) \] \[ \sin(150^\circ) = \sin(30^\circ) = \frac{1}{2} \] \[ v_y = \frac{9}{7} \cdot \frac{1}{2} = \frac{9}{14} \] Thus, the component form of vector \(\mathbf{v}\) is: \[ \mathbf{v} = \left( -\frac{9\sqrt{3}}{14}, \frac{9}{14} \right) \] To visualize \(\mathbf{v}\), plot a vector in the coordinate plane starting from the origin \((0,0)\) with the tip at \(\left( -\frac{9\sqrt{3}}{14}, \frac{9}{14}
Transcribed Image Text:**Find the component form of v given its magnitude and the angle it makes with the positive x-axis.** **Magnitude:** \[ \|v\| = \frac{9}{7} \] **Angle:** \[ \theta = 150^\circ \] \[ \text{Sketch } \mathbf{v}: \] (Insert the sketch of vector \(\mathbf{v}\) here) --- To find the component form of vector \(\mathbf{v}\), we use the magnitude \(\|v\|\) and the angle \(\theta\) it makes with the positive x-axis. The components of \(\mathbf{v}\) can be calculated using the following formulas: \[ v_x = \|v\| \cos(\theta) \] \[ v_y = \|v\| \sin(\theta) \] Given: \[ \|v\| = \frac{9}{7} \] \[ \theta = 150^\circ \] We calculate the components as follows: **For the x-component:** \[ v_x = \frac{9}{7} \cos(150^\circ) \] \[ \cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} \] \[ v_x = \frac{9}{7} \cdot -\frac{\sqrt{3}}{2} = -\frac{9\sqrt{3}}{14} \] **For the y-component:** \[ v_y = \frac{9}{7} \sin(150^\circ) \] \[ \sin(150^\circ) = \sin(30^\circ) = \frac{1}{2} \] \[ v_y = \frac{9}{7} \cdot \frac{1}{2} = \frac{9}{14} \] Thus, the component form of vector \(\mathbf{v}\) is: \[ \mathbf{v} = \left( -\frac{9\sqrt{3}}{14}, \frac{9}{14} \right) \] To visualize \(\mathbf{v}\), plot a vector in the coordinate plane starting from the origin \((0,0)\) with the tip at \(\left( -\frac{9\sqrt{3}}{14}, \frac{9}{14}
The image contains four coordinate planes each labeled with the x and y axes, illustrating vectors and angles in standard position. Here is a detailed breakdown of each graph: 1. **Top-Left Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originating from the origin (0,0) extends into the second quadrant. - The vector creates an angle of 150 degrees with the positive x-axis, measured counterclockwise. - The angle is highlighted by a red arc indicating the 150-degree measurement. 2. **Top-Right Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originates from the origin (0,0) and extends into the third quadrant. - The vector creates an interior angle of 150 degrees with the positive y-axis, measured clockwise. - The 150-degree angle is indicated by a red arc. 3. **Bottom-Left Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originates from the origin (0,0) and extends into the first quadrant. - The vector creates an angle of 150 degrees with the positive x-axis, measured counterclockwise. - This angle is shown with a red arc indicating the 150-degree measurement. 4. **Bottom-Right Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originates from the origin (0,0) and extends into the fourth quadrant. - The vector forms an interior angle of 150 degrees with the negative x-axis, measured clockwise. - The angle is highlighted by a red arc indicating the 150-degree measurement. Each graph visualizes how the angle of 150 degrees can be represented through vectors in different quadrants of the coordinate plane, showcasing the concepts of angle orientation and vector direction.
Transcribed Image Text:The image contains four coordinate planes each labeled with the x and y axes, illustrating vectors and angles in standard position. Here is a detailed breakdown of each graph: 1. **Top-Left Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originating from the origin (0,0) extends into the second quadrant. - The vector creates an angle of 150 degrees with the positive x-axis, measured counterclockwise. - The angle is highlighted by a red arc indicating the 150-degree measurement. 2. **Top-Right Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originates from the origin (0,0) and extends into the third quadrant. - The vector creates an interior angle of 150 degrees with the positive y-axis, measured clockwise. - The 150-degree angle is indicated by a red arc. 3. **Bottom-Left Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originates from the origin (0,0) and extends into the first quadrant. - The vector creates an angle of 150 degrees with the positive x-axis, measured counterclockwise. - This angle is shown with a red arc indicating the 150-degree measurement. 4. **Bottom-Right Coordinate Plane:** - The x- and y-axes both range from -1.5 to 1.5. - A vector originates from the origin (0,0) and extends into the fourth quadrant. - The vector forms an interior angle of 150 degrees with the negative x-axis, measured clockwise. - The angle is highlighted by a red arc indicating the 150-degree measurement. Each graph visualizes how the angle of 150 degrees can be represented through vectors in different quadrants of the coordinate plane, showcasing the concepts of angle orientation and vector direction.
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