Consider the vectors shown in the figure.

### Part 1: Basic properties

The figure shows vectors in $\mathbb{R}^n$ for $n =$ . The coordinate representation of the zero vector is $\vec{0} =$ (use coordinate vector notation, not ijk-vector notation).

### Part 2: Parallel or not?

Are any two of these vectors parallel to each other?
• Is $\vec{a}_1$ parallel to $\vec{a}_2$?
• Is $\vec{a}_1$ parallel to $\vec{a}_3$?
• Is $\vec{a}_2$ parallel to $\vec{a}_3$?

### Part 3: Coordinate representations

Find the coordinate representations of each vector:
• $\vec{a}_1 =$
• $\vec{a}_2 =$
• $\vec{a}_3 =$

### Part 4: Finding relationships among these vectors

Write each of the vectors in terms of the other vectors. Use the names of vectors in your answer (e.g., enter 4a2 - 5a3 for $4 \vec{a}_2 - 5 \vec{a}_3$). Do not enter coordinate representations such as $\langle 3, 4 \rangle$.
• $\vec{a}_1 =$
• $\vec{a}_2 =$
• $\vec{a}_3 =$

### Part 5: Can we combine these vectors to get the zero vector?

If possible, scale and add the vectors $\vec{a}_1$, $\vec{a}_2$, and $\vec{a}_3$ to obtain the zero vector $\vec{0}$. If this is not possible, enter DNE.
Use the names of vectors in your answer (e.g., enter 4a1 - 5a2 + a3 for $4 \vec{a}_1 - 5 \vec{a}_2 + \vec{a}_3$). Do not enter coordinate representations such as $\langle 3, 4 \rangle$.
$= \vec{0}.$