Chapter 1: Problems (1.82 - 1.86)

1.82 For the two vectors $\vec{A}$ and $\vec{B}$ in Fig. 1.30, a) find the scalar product $\vec{A} \cdot \vec{B}$; b) find the magnitude and direction of the vector product $\vec{A} \times \vec{B}$.

1.83 Figure 1.8c shows a parallelogram based on the two vectors $\vec{A}$ and $\vec{B}$. a) Show that the magnitude of the cross product of these two vectors is equal to the area of the parallelogram. (Hint: area = base $\times$ height.) b) What is the angle between the cross product and the plane of the parallelogram?

1.84 The vector $\vec{A}$ is 3.50 cm long and is directed into this page. Vector $\vec{B}$ points from the lower-right corner of this page to the upper-left corner of this page. Define an appropriate right-handed coordinate system and find the three components of the vector product $\vec{A} \times \vec{B}$, measured in cm$^2$. In a diagram, show your coordinate system and the vectors $\vec{A}$, $\vec{B}$, and $\vec{A} \times \vec{B}$.

1.85 Given two vectors $\vec{A} = -2.00\hat{i} + 3.00\hat{j} + 4.00\hat{k}$ and $\vec{B} = 3.00\hat{i} + 1.00\hat{j} - 3.00\hat{k}$, do the following. a) Find the magnitude of each vector. b) Write an expression for the vector difference $\vec{A} - \vec{B}$. Is this the same as the magnitude of $\vec{B} - \vec{A}$? Explain.

1.86 Bond Angle in Methane. In the methane molecule, CH$_4$, each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates where one of the C – H bonds is in the direction of $\hat{i} + \hat{j} + \hat{k}$, an adjacent C – H bond is in the direction of $\hat{i} - \hat{j} - \hat{k}$ direction. Calculate the angle between these two bonds.