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ition vector v is in quadrant II, then to find the direction angle, use the following formula: 0 = tan-¹ O False -180

ition vector v is in quadrant II, then to find the direction angle, use the following formula: 0 = tan-¹ O False -180

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter1: The Six Trigonometric Functions
Section1.2: The Rectangular Coordinate System
Problem 92PS: Draw an angle in standard position whose terminal side contains the point (2, –3). Find the...
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### Understanding the Direction Angle of a Vector **Question:** If the terminating point of a position vector \( \mathbf{v} \) is in quadrant II, then to find the direction angle, use the following formula: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) + 180^\circ \). - True - False --- **Concept Explanation:** For a position vector terminating in quadrant II, we need to consider the characteristics of that quadrant. Quadrant II is where the x-coordinate (abscissa) is negative and the y-coordinate (ordinate) is positive. The direction angle \( \theta \) is typically measured counterclockwise from the positive x-axis. When dealing with vectors in quadrant II, the tangent function, which is defined as the ratio of the y-coordinate to the x-coordinate (\( \tan(\theta) = \frac{y}{x} \)), results in a negative ratio since \( x \) is negative and \( y \) is positive. To correct this and to find the correct angle in the context of the second quadrant, you add \( 180^\circ \) to the angle obtained from the \( \tan^{-1} \left( \frac{y}{x} \right) \) function, ensuring you are finding the correct direction angle that places the vector accurately in the plane geometry. #### Answer: - True --- **Visual Representation:** Imagine a standard Cartesian coordinate system divided into four quadrants. The second quadrant is located to the top-left part of the system, where values of \( x \) are negative and values of \( y \) are positive. This impacts how the direction angle is calculated and interpreted. For comprehensive understanding, study the use of the arctangent function and its implications in different quadrants while solving vector-related problems.
Transcribed Image Text:### Understanding the Direction Angle of a Vector **Question:** If the terminating point of a position vector \( \mathbf{v} \) is in quadrant II, then to find the direction angle, use the following formula: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) + 180^\circ \). - True - False --- **Concept Explanation:** For a position vector terminating in quadrant II, we need to consider the characteristics of that quadrant. Quadrant II is where the x-coordinate (abscissa) is negative and the y-coordinate (ordinate) is positive. The direction angle \( \theta \) is typically measured counterclockwise from the positive x-axis. When dealing with vectors in quadrant II, the tangent function, which is defined as the ratio of the y-coordinate to the x-coordinate (\( \tan(\theta) = \frac{y}{x} \)), results in a negative ratio since \( x \) is negative and \( y \) is positive. To correct this and to find the correct angle in the context of the second quadrant, you add \( 180^\circ \) to the angle obtained from the \( \tan^{-1} \left( \frac{y}{x} \right) \) function, ensuring you are finding the correct direction angle that places the vector accurately in the plane geometry. #### Answer: - True --- **Visual Representation:** Imagine a standard Cartesian coordinate system divided into four quadrants. The second quadrant is located to the top-left part of the system, where values of \( x \) are negative and values of \( y \) are positive. This impacts how the direction angle is calculated and interpreted. For comprehensive understanding, study the use of the arctangent function and its implications in different quadrants while solving vector-related problems.
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