Three vectors a = 3i+4j, b = 6i+8j, c = -6i-8c. Identify which are (i) equal (ii) parallel (iii) opposite.

Two vectors are equal if their corresponding components are identical.
Comparing the components:
$\vec{a} = (3, 4)$
$\vec{b} = (-6, 8)$
$\vec{c} = (-6, -8)$
No two vectors have identical components.
Therefore, no two vectors are equal.

Two vectors $\vec{u}$ and $\vec{v}$ are parallel if $\vec{v} = k\vec{u}$ for some non-zero scalar $k$. This means their corresponding components must be proportional.

• Check $\vec{a}$ and $\vec{b}$: Assume $\vec{b} = k\vec{a}$:

  $$-6\vec{i} + 8\vec{j} = k(3\vec{i} + 4\vec{j})$$ $$-6\vec{i} + 8\vec{j} = 3k\vec{i} + 4k\vec{j}$$

  Equating the components:

  $$-6 = 3k \implies k = -2$$ $$8 = 4k \implies k = 2$$

  Since the value of $k$ is not consistent ($-2 \neq 2$), $\vec{a}$ and $\vec{b}$ are not parallel.

• Check $\vec{a}$ and $\vec{c}$: Assume $\vec{c} = k\vec{a}$:

  $$-6\vec{i} - 8\vec{j} = k(3\vec{i} + 4\vec{j})$$ $$-6\vec{i} - 8\vec{j} = 3k\vec{i} + 4k\vec{j}$$

  Equating the components:

  $$-6 = 3k \implies k = -2$$ $$-8 = 4k \implies k = -2$$

  Since $k = -2$ is consistent for both components, $\vec{a}$ and $\vec{c}$ are parallel.

• Check $\vec{b}$ and $\vec{c}$: Assume $\vec{c} = k\vec{b}$:

  $$-6\vec{i} - 8\vec{j} = k(-6\vec{i} + 8\vec{j})$$ $$-6\vec{i} - 8\vec{j} = -6k\vec{i} + 8k\vec{j}$$

  Equating the components:

  $$-6 = -6k \implies k = 1$$ $$-8 = 8k \implies k = -1$$

  Since the value of $k$ is not consistent ($1 \neq -1$), $\vec{b}$ and $\vec{c}$ are not parallel.

Therefore, the parallel vectors are $\boxed{\vec{a} \text{ and } \vec{c}}$.

Strictly, two vectors $\vec{u}$ and $\vec{v}$ are opposite if $\vec{v} = -\vec{u}$. This means they have the same magnitude and point in exactly opposite directions.
First, calculate the magnitudes of the vectors:

For vectors to be opposite, their magnitudes must be equal. Only $\vec{b}$ and $\vec{c}$ have equal magnitudes.
Let's check if $\vec{c} = -\vec{b}$:

Since $\vec{c} = -6\vec{i} - 8\vec{j}$ and $-\vec{b} = 6\vec{i} - 8\vec{j}$, they are not equal. Thus, $\vec{b}$ and $\vec{c}$ are not opposite.
Under the strict definition, no two vectors are opposite.

However, "opposite" can also be interpreted as "parallel and pointing in the opposite direction" (i.e., $\vec{v} = k\vec{u}$ where $k < 0$).
From Step 2, we found that $\vec{a}$ and $\vec{c}$ are parallel, with $\vec{c} = -2\vec{a}$. Since $k = -2$ (which is negative), $\vec{a}$ and $\vec{c}$ are parallel and point in opposite directions.
Therefore, the vectors that are parallel and point in opposite directions are $\boxed{\vec{a} \text{ and } \vec{c}}$.