Problem 79
Vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\) have scalar product \(-\)6.00, and their vector product has magnitude \(+\)9.00. What is the angle between these two vectors?
Problem 80
A cube is placed so that one corner is at the origin and three edges are along the \(x\)-, \(y\)-, and \(z\)-axes of a coordinate system (Fig. P1.80). Use vectors to compute (a) the angle between the edge along the z axis (line \(ab\)) and the diagonal from the origin to the opposite corner (line \(ad\)), and (b) the angle between line \(ac\) (the diagonal of a face) and line ad.
Problem 81
Vector \(\overrightarrow{A}\) has magnitude 12.0 m, and vector \(\overrightarrow{B}\) has magnitude 16.0 m. The scalar product \(\overrightarrow{A}\) \(\cdot\) \(\overrightarrow{B}\) is 112.0 m\(^2\). What is the magnitude of the vector product between these two vectors?
Problem 83
The scalar product of vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\) is \(+\)48.0 m\(^2\). Vector \(\overrightarrow{A}\) has magnitude 9.00 m and direction 28.0\(^{\circ}\) west of south. If vector \(\overrightarrow{B}\) has direction 39.0\(^{\circ}\) south of east, what is the magnitude of \(\overrightarrow{B}\)?
Problem 84
Two vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\) have magnitudes \(A\) = 3.00 and \(B\) = 3.00. Their vector product is \(\overrightarrow{A}\) \\(\times\\) \(\overrightarrow{B}$$^{\circ}\) = \(-\)5.00\(\hat{k}\) + 2.00\(\hat{\imath}\). What is the angle between \(\overrightarrow{A}\) and \(\overrightarrow{B}\)?
Problem 86
Later in our study of physics we will encounter quantities represented by (\(\overrightarrow{A}\) \\(\times\\) \(\overrightarrow{B}\)) \(\cdot\) \(\overrightarrow{C}\). (a) Prove that for any three vectors \(\overrightarrow{A}\), \(\overrightarrow{B}\), and \(\overrightarrow{C}\), \(\overrightarrow{A}\) \(\cdot\) (\(\overrightarrow{B}\) \\(\times\\) \(\overrightarrow{C}\)) = (\(\overrightarrow{A}\) \\(\times\\) \(\overrightarrow{B}\)) \(\cdot\) \(\overrightarrow{C}\). (b) Calculate (\(\overrightarrow{A}\) \\(\times\\) \(\overrightarrow{B}\)) \(\cdot\) \(\overrightarrow{C}\) for vector \(\overrightarrow{A}\) with magnitude \(A\) = 5.00 and angle \(\theta$$_A\) = 26.0\(^{\circ}\) (measured from the +\(x\)-axis toward the \(+$$y\)-axis), vector \(\overrightarrow{B}\) with \(B\) = 4.00 and \(\theta$$_B\) = 63.0\(^{\circ}\), and vector \(\overrightarrow{C}\) with magnitude 6.00 and in the \(+$$z\)-direction. Vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\) are in the \(x$$y\)-plane.
