Chapter 1: Exercises (1.46 - 1.50)

1.46 a) Write each vector in Fig. 1.30 in terms of the unit vectors $\hat{i}$ and $\hat{j}$. b) Use unit vectors to express the vector $\vec{C}$, where $\vec{C} = 3.00\vec{A} - 4.00\vec{B}$. c) Find the magnitude and direction of $\vec{C}$.

1.47 Given two vectors $\vec{A} = 4.00\hat{i} + 3.00\hat{j}$ and $\vec{B} = 5.00\hat{i} - 2.00\hat{j}$, a) find the magnitude of each vector; b) write an expression for the vector difference $\vec{A}-\vec{B}$ using unit vectors; c) find the magnitude and direction of the vector difference $\vec{A}-\vec{B}$. d) In a vector diagram show $\vec{A}, \vec{B}$ and $\vec{A}-\vec{B}$ and also show that your diagram agrees qualitatively with your answer in part (c).

1.48 a) Is the vector $(\hat{i} +  \hat{j} + \hat{k})$ a unit vector? Justify your answer. b) Can a unit vector have any components with a magnitude greater than unity? Can it have any negative components? In each case justify your answer. c) If $\vec{A} = a(3.0\hat{i} + 4.0\vec{j})$, where $a$ is a constant, determine the value of $a$ that makes $\vec{A}$ a unit vector.

1.49 a) Use vector components to prove that two vectors commute for both addition and the scalar product. b) Use vector components to prove that two vectors anticommute for the vector product. That is, prove that $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$.

Section 1.10 Products of Vectors

1.50 For the vectors $\vec{A}, \vec{B},$ and $\vec{C}$ in Fig. 1.28, find the scalar products a) $\vec{A} \cdot \vec{B}$; b) $\vec{B} \cdot \vec{C}$; c) $\vec{A} \cdot \vec{C}$.