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Later in our study of physics we will encounter quantities
Chapter 1, Problem 97P(choose chapter or problem)
Later in our study of physics we will encounter quantities represented by \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\).
(a) Prove that for any three vectors \(\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}}\), and \(\overrightarrow{\boldsymbol{C}}, \overrightarrow{\boldsymbol{A}} \cdot(\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{C}})=(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\).
(b) Calculate \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) for the three vectors \(\overrightarrow{\boldsymbol{A}}\) with magnitude A = 5.00 and angle \(\theta_A=26.0^{\circ}\) measured in the sense from the +x-axis toward the +y-axis, \(\overrightarrow{\boldsymbol{B}}\) with B = 4.00 and \(\theta_B=63.0^{\circ}\), and \(\overrightarrow{\boldsymbol{C}}\) with magnitude 6.00 and in the +z-direction. Vectors \(\vec{A}\) and \(\vec{B}\) are in the xy-plane.
Questions & Answers
QUESTION:
Later in our study of physics we will encounter quantities represented by \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\).
(a) Prove that for any three vectors \(\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}}\), and \(\overrightarrow{\boldsymbol{C}}, \overrightarrow{\boldsymbol{A}} \cdot(\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{C}})=(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\).
(b) Calculate \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) for the three vectors \(\overrightarrow{\boldsymbol{A}}\) with magnitude A = 5.00 and angle \(\theta_A=26.0^{\circ}\) measured in the sense from the +x-axis toward the +y-axis, \(\overrightarrow{\boldsymbol{B}}\) with B = 4.00 and \(\theta_B=63.0^{\circ}\), and \(\overrightarrow{\boldsymbol{C}}\) with magnitude 6.00 and in the +z-direction. Vectors \(\vec{A}\) and \(\vec{B}\) are in the xy-plane.
ANSWER:Solution 97P a) The thing which has to be proved is, A.(B × C) = (A × B).C The left hand side can be shown as, The right hand side can be shown as, This shows the determinant of the transpose of the previous matrix. As, we know that the