The problem asks to identify which of the given conditions imply that vector $a$ is parallel to vector $b$. The conditions are: 1. $a = b$

1. Problem Description

The problem asks to identify which of the given conditions imply that vector $a$ is parallel to vector $b$. The conditions are:

1. $a = b$

2. $|a| = |b|$

3. $a$ and $b$ have opposite directions

4. $|a| = 0$ or $|b| = 0$

5. $a$ and $b$ are unit vectors

2. Solution Steps

Two vectors are parallel if one is a scalar multiple of the other. This means $a = kb$ for some scalar $k$.

Condition 1: $a = b$. This implies $a = 1 \cdot b$, so $a$ is parallel to $b$.

Condition 2: $|a| = |b|$. This only means the magnitudes are equal. For example, $a = (1, 0)$ and $b = (0, 1)$ have the same magnitude but are not parallel.

Condition 3: $a$ and $b$ have opposite directions. This implies $a = -k \cdot b$ for some $k > 0$. Therefore $a$ is parallel to $b$.

Condition 4: $|a| = 0$ or $|b| = 0$. If $|a| = 0$, then $a = (0, 0)$. If $|b| = 0$, then $b = (0, 0)$. If $a = 0$ and $b$ is any vector, then $a = 0 \cdot b$ so $a$ and $b$ are parallel. Similarly, if $b=0$ and $a$ is any vector, then $b = 0\cdot a$ so $b$ and $a$ are parallel. If both $a$ and $b$ are the zero vector, they are parallel since $a = 1 \cdot b$.

Condition 5: $a$ and $b$ are unit vectors. This only means $|a| = 1$ and $|b| = 1$. Like condition 2, they can have different directions. For example, $a=(1,0)$ and $b=(0,1)$ are unit vectors, but are not parallel.

The conditions that ensure $a$ and $b$ are parallel are 1, 3, and

4.

3. Final Answer

134