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69. For the vectors in the earlier figure, find (a) (A × F ). D, 6) (A × F )·(B x B ), and (c) ( A F)(D x B).

69. For the vectors in the earlier figure, find (a) (A × F ). D, 6) (A × F )·(B x B ), and (c) ( A F)(D x B).

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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#69, using the figure above #53.

**Vectors and Their Properties** 69. For the vectors in the earlier figure, find: (a) \(( \vec{A} \times \vec{F} ) \cdot \vec{D}\), (b) \(( \vec{A} \times \vec{F} ) \cdot ( \vec{D} \times \vec{B} )\), (c) \(( \vec{A} \cdot \vec{F} )( \vec{D} \times \vec{B} )\). 70. Answer the following: (a) If \(\vec{A} \times \vec{F} = \vec{B} \times \vec{F}\), can we conclude \(\vec{A} = \vec{B}\)? (b) If \(\vec{A} \cdot \vec{F} = \vec{B} \cdot \vec{F}\), can we conclude \(\vec{A} = \vec{B}\)? (c) If \(F \vec{A} = \vec{B} F\), can we conclude \(\vec{A} = \vec{B}\)? Why or why not? **Conversions and Distances** nmi = 1852 m. 76. An air traffic controller notices two signals from two planes on the radar monitor. - One plane is at an altitude of 800 m and in a 19.2-km horizontal distance to the tower in a direction \(25^\circ\) south of west. - The second plane is at an altitude of 1100 m and its horizontal distance is 17.6 km and \(20^\circ\) south of west. What is the distance between these planes? **Vector Addition and Trigonometry** 77. Show that when \(\vec{A} + \vec{B} = \vec{C}\), then \(C^2 = A^2 + B^2 + 2AB \cos \varphi\), where \(\varphi\) is the angle between vectors \(\vec{A}\) and \(\vec{B}\).
Transcribed Image Text:**Vectors and Their Properties** 69. For the vectors in the earlier figure, find: (a) \(( \vec{A} \times \vec{F} ) \cdot \vec{D}\), (b) \(( \vec{A} \times \vec{F} ) \cdot ( \vec{D} \times \vec{B} )\), (c) \(( \vec{A} \cdot \vec{F} )( \vec{D} \times \vec{B} )\). 70. Answer the following: (a) If \(\vec{A} \times \vec{F} = \vec{B} \times \vec{F}\), can we conclude \(\vec{A} = \vec{B}\)? (b) If \(\vec{A} \cdot \vec{F} = \vec{B} \cdot \vec{F}\), can we conclude \(\vec{A} = \vec{B}\)? (c) If \(F \vec{A} = \vec{B} F\), can we conclude \(\vec{A} = \vec{B}\)? Why or why not? **Conversions and Distances** nmi = 1852 m. 76. An air traffic controller notices two signals from two planes on the radar monitor. - One plane is at an altitude of 800 m and in a 19.2-km horizontal distance to the tower in a direction \(25^\circ\) south of west. - The second plane is at an altitude of 1100 m and its horizontal distance is 17.6 km and \(20^\circ\) south of west. What is the distance between these planes? **Vector Addition and Trigonometry** 77. Show that when \(\vec{A} + \vec{B} = \vec{C}\), then \(C^2 = A^2 + B^2 + 2AB \cos \varphi\), where \(\varphi\) is the angle between vectors \(\vec{A}\) and \(\vec{B}\).
**Figure 2.33: Vector Representation and Angles** This figure illustrates several vectors labeled \(\vec{A}\), \(\vec{B}\), \(\vec{C}\), \(\vec{D}\), and \(\vec{F}\), each with a specified magnitude and angle of orientation relative to a horizontal reference line. 1. **Coordinate Axes:** - A pair of perpendicular axes, labeled \(x\) (horizontal) and \(y\) (vertical). 2. **Vector \(\vec{A}\):** - Magnitude: \(A = 10.0\) - Angle: \(30^\circ\) above the horizontal axis. 3. **Vector \(\vec{B}\):** - Magnitude: \(B = 5.0\) - Angle: \(53^\circ\) above the horizontal axis. 4. **Vector \(\vec{C}\):** - Magnitude: \(C = 12.0\) - Angle: \(60^\circ\) below the horizontal axis. 5. **Vector \(\vec{D}\):** - Magnitude: \(D = 20.0\) - Angle: \(37^\circ\) below the horizontal axis. 6. **Vector \(\vec{F}\):** - Magnitude: \(F = 20.0\) - Angle: \(30^\circ\) above the horizontal axis. **Exercise 53: Vector Equation Analysis** **Problem Statement:** Given the vectors in Figure 2.33, determine the vector \(\vec{R}\) that solves the following equations: (a) \(\vec{D} + \vec{R} = \vec{F}\) (b) \(\vec{C} - 2\vec{D} + 5\vec{R} = 3\vec{F}\) **Assumptions:** - The \(+x\)-axis is assumed to be horizontal, directed to the right. These exercises require the application of vector addition and subtraction principles, considering both the magnitudes and directional angles of the given vectors.
Transcribed Image Text:**Figure 2.33: Vector Representation and Angles** This figure illustrates several vectors labeled \(\vec{A}\), \(\vec{B}\), \(\vec{C}\), \(\vec{D}\), and \(\vec{F}\), each with a specified magnitude and angle of orientation relative to a horizontal reference line. 1. **Coordinate Axes:** - A pair of perpendicular axes, labeled \(x\) (horizontal) and \(y\) (vertical). 2. **Vector \(\vec{A}\):** - Magnitude: \(A = 10.0\) - Angle: \(30^\circ\) above the horizontal axis. 3. **Vector \(\vec{B}\):** - Magnitude: \(B = 5.0\) - Angle: \(53^\circ\) above the horizontal axis. 4. **Vector \(\vec{C}\):** - Magnitude: \(C = 12.0\) - Angle: \(60^\circ\) below the horizontal axis. 5. **Vector \(\vec{D}\):** - Magnitude: \(D = 20.0\) - Angle: \(37^\circ\) below the horizontal axis. 6. **Vector \(\vec{F}\):** - Magnitude: \(F = 20.0\) - Angle: \(30^\circ\) above the horizontal axis. **Exercise 53: Vector Equation Analysis** **Problem Statement:** Given the vectors in Figure 2.33, determine the vector \(\vec{R}\) that solves the following equations: (a) \(\vec{D} + \vec{R} = \vec{F}\) (b) \(\vec{C} - 2\vec{D} + 5\vec{R} = 3\vec{F}\) **Assumptions:** - The \(+x\)-axis is assumed to be horizontal, directed to the right. These exercises require the application of vector addition and subtraction principles, considering both the magnitudes and directional angles of the given vectors.
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