To determine if the two vectors $\frac{1}{3}a + \frac{2}{3}b$ and $\frac{2}{3}a + \frac{4}{3}b$ are parallel, we can check if one is a scalar multiple of the other.

Step 1: Express the vectors in simpler form.

The two vectors are:

1. $v_1 = \frac{1}{3}a + \frac{2}{3}b$
2. $v_2 = \frac{2}{3}a + \frac{4}{3}b$

These can be rewritten as:

1. $v_1 = \left( \frac{1}{3}, \frac{2}{3} \right)$
2. $v_2 = \left( \frac{2}{3}, \frac{4}{3} \right)$

Step 2: Check if $v_2$ is a scalar multiple of $v_1$.

For the vectors to be parallel, we must have: $v_2 = k \cdot v_1$ for some scalar $k$.

Comparing the two components of $v_2$ and $v_1$:

1. For the first component: $\frac{2}{3} = k \cdot \frac{1}{3}$ ⟹ $k = 2$
2. For the second component: $\frac{4}{3} = k \cdot \frac{2}{3}$ ⟹ $k = 2$

Since $k = 2$ is consistent for both components, the vectors are scalar multiples of each other.

Conclusion:

The vectors $\frac{1}{3}a + \frac{2}{3}b$ and $\frac{2}{3}a + \frac{4}{3}b$ are parallel.

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Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do you find if 3D vectors are parallel?
2. Can two zero vectors be considered parallel?
3. How can the cross product be used to check for parallelism?
4. What is the geometric interpretation of parallel vectors?
5. How would you check if two lines are parallel in a coordinate plane?

Tip: When checking if vectors are parallel, one way is to check if their components are proportional.