Let's solve the given problem step by step.

Part (a) - Finding two unit vectors parallel to $v = 4i + 3j$

A unit vector is a vector with magnitude 1. To find a unit vector in the direction of $v = 4i + 3j$, we first find the magnitude of $v$:

$\| v \| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Now, the unit vector in the same direction as $v$ is given by:

$\hat{v} = \frac{v}{\| v \|} = \frac{4i + 3j}{5} = \left(\frac{4}{5}\right)i + \left(\frac{3}{5}\right)j$

Thus, the parallel unit vector with the same direction is:

$\boxed{\left(\frac{4}{5}, \frac{3}{5}\right)}$

Additionally, there is another unit vector parallel but in the opposite direction, which is:

$-\hat{v} = -\left(\frac{4}{5}i + \frac{3}{5}j\right) = \left(\frac{-4}{5}, \frac{-3}{5}\right)$

Thus, the two unit vectors parallel to $v$ are:

$\boxed{\left(\frac{4}{5}, \frac{3}{5}\right)} \quad \text{and} \quad \boxed{\left(\frac{-4}{5}, \frac{-3}{5}\right)}$

Part (b) - Finding $b$ if $v = \frac{1}{8} b$ is a unit vector

Given that $v = \frac{1}{8} b$ is a unit vector, the magnitude of $v$ must be 1. Since $v = \frac{1}{8} b$, we have:

$\left\| \frac{1}{8} b \right\| = 1$

This implies:

$\frac{1}{8} \| b \| = 1$

Multiplying both sides by 8, we get:

$\| b \| = 8$

Therefore, the magnitude of $b$ is 8.

Part (c) - Finding all values of $a$ such that $w = ai - \frac{a}{3} j$ is a unit vector

We are given $w = ai - \frac{a}{3} j$ and need to find all values of $a$ such that $w$ is a unit vector. To do this, we calculate the magnitude of $w$:

$\| w \| = \sqrt{a^2 + \left( \frac{a}{3} \right)^2} = \sqrt{a^2 + \frac{a^2}{9}} = \sqrt{\frac{9a^2}{9} + \frac{a^2}{9}} = \sqrt{\frac{10a^2}{9}} = \frac{\sqrt{10} |a|}{3}$

For $w$ to be a unit vector, $\| w \| = 1$, so we set:

$\frac{\sqrt{10} |a|}{3} = 1$

Multiplying both sides by 3:

$\sqrt{10} |a| = 3$

Dividing both sides by $\sqrt{10}$:

$|a| = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}$

Thus, $a = \pm \frac{3\sqrt{10}}{10}$.

Summary of Answers:

1. (a) The two unit vectors parallel to $v$ are $\left( \frac{4}{5}, \frac{3}{5} \right)$ and $\left( \frac{-4}{5}, \frac{-3}{5} \right)$.
2. (b) The magnitude of $b$ is 8.
3. (c) The values of $a$ such that $w$ is a unit vector are $a = \pm \frac{3\sqrt{10}}{10}$.

Let me know if you want further details on any of the steps!

Follow-up Questions:

1. What is a unit vector, and why is its magnitude always 1?
2. How do you find the direction of a vector if you know its components?
3. What does it mean for two vectors to be parallel?
4. How does scaling a vector affect its magnitude and direction?
5. How can you generalize the process of finding a unit vector for any given vector?

Tip:

Always verify if a vector is a unit vector by ensuring its magnitude equals 1.