Chapter 1: Discussion Questions (Q1.21 - Q1.23)

Q1.21 Let $\bf \overrightarrow{A}$ represent any nonzero vector. Why is $\bf \overrightarrow{A}$/A a unit vector and what is it's direction? If $\theta$ is the angle that $\bf \overrightarrow{A}$ makes with the $+x$-axis, explain why ($\bf \overrightarrow{A}$/A)$\cdot\hat{i}$ is called the direction cosine for that axis.

Response: By definition, a unit vector is vector with a magnitude of 1 and no units. Its only purpose is to point; that is, to describe a direction in space. $\bf \overrightarrow{A}$/A is a vector with a magnitude of 1 and no units (the units of the vector cancel out with the units of the magnitude). The direction of that vector is the same as $\bf \overrightarrow{A}$. ($\bf \overrightarrow{A}$/A)$\cdot\hat{i} = (1)(1)cos\theta = cos\theta$.


Q1.22 Which of the following are legitimate mathematical operations:
a) $\bf \overrightarrow{A}\cdot(\bf \overrightarrow{B}-\bf \overrightarrow{C})$;
b) $(\bf \overrightarrow{A}- \bf \overrightarrow{B})\times\bf \overrightarrow{C}$;
c) $\bf \overrightarrow{A}\cdot(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$;
d) $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$;
e) $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\cdot\bf \overrightarrow{C})$?
In each case, give the reason for your answer.

Response:
a) $(\bf \overrightarrow{B}-\bf \overrightarrow{C})$ is a vector quantity, so this is a valid operation.
b) $(\bf \overrightarrow{A}- \bf \overrightarrow{B})$ is a vector quantity, so this is a valid operation.
c) $(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$ is a vector quantity, so this is valid operation.
d) See (c).
e) $(\bf \overrightarrow{B}\cdot\bf \overrightarrow{C})$ is a scalar quantity, so this is not a valid operation.


Q1.23 Consider the two repeated vector products $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$ and $(\bf \overrightarrow{A}\times\bf \overrightarrow{B})\times\bf \overrightarrow{C}$. Give an example that illustrates the general rule that these two vector products do not have the same magnitude or direction. Can you choose vectors $\bf \overrightarrow{A}$, $\bf \overrightarrow{B}$, and $\bf \overrightarrow{C}$ such that these two vector products are equal to each other? If so, give an example.

Response: Take the vectors
$\bf \overrightarrow{A} = 1\hat{i} + 2\hat{j} + 0\hat{k}$
$\bf \overrightarrow{B} = 2\hat{i} + 1\hat{j} + 0\hat{k}$ and
$\bf \overrightarrow{C} = 3\hat{i} + 4\hat{j} + 0\hat{k}$

Let $\bf \overrightarrow{D} = (\bf \overrightarrow{B}\times\bf \overrightarrow{C})$. Then $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C}) = \bf \overrightarrow{A}\times\overrightarrow{D} =  \overrightarrow{E}$

Calculate the vector product $\bf \overrightarrow{D} =  0\hat{i} + 0\hat{j} + 5\hat{k}$ Then $\bf \overrightarrow{A}\times\overrightarrow{D} =  \overrightarrow{E} = 10\hat{i} - 5\hat{j} + 0\hat{k}$

Let $\bf \overrightarrow{F} = (\bf \overrightarrow{A}\times\bf \overrightarrow{B})$. Then $(\bf \overrightarrow{A}\times(\bf \overrightarrow{B})\times\bf \overrightarrow{C}) = \bf \overrightarrow{F}\times\overrightarrow{C} =  \overrightarrow{G}$

Calculate the vector product $\bf \overrightarrow{F} =  0\hat{i} + 0\hat{j} - 3\hat{k}$ Then $\bf \overrightarrow{F}\times\overrightarrow{C} =  \overrightarrow{G} = 12\hat{i} - 9\hat{j} + 0\hat{k}$

Clearly, $\bf \overrightarrow{E}\neq \overrightarrow{G}$

It is possible to choose three distinct nonzero vectors where $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C}) = (\bf \overrightarrow{A}\times\bf \overrightarrow{B})\times\bf \overrightarrow{C}$

Choose the unit vectors $\hat{i}, \hat{j},$ and $\hat{k}$. Since $\hat{j}\times\hat{k}=\hat{i}$, and $\hat{i}\times\hat{j}=\hat{k}$,
$\hat{i}\times(\hat{j}\times\hat{k}) = \hat{i}\times\hat{i} = 0$ and $(\hat{i}\times\hat{j})\times\hat{k} = \hat{k}\times\hat{k} = 0$
$$\hat{i}\times(\hat{j}\times\hat{k}) = (\hat{i}\times\hat{j})\times\hat{k}$$