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

Here are the solutions for the vector problem.

The given vectors are: $\vec{a} = 3\vec{i} + 4\vec{j}$ $\vec{b} = -6\vec{i} + 8\vec{j}$ $\vec{c} = -6\vec{i} - 8\vec{j}$

Step 1: Identify equal vectors. 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.

Step 2: Identify parallel vectors. 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 $\boxed{\vec{a} and \vec{c}}$.

Step 3: Identify opposite vectors. 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: $|\vec{a}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ $|\vec{b}| = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ $|\vec{c}| = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ For vectors to be opposite, their magnitudes must be equal. Only $\vec{b}$ and $\vec{c}$ have equal magnitudes. Let's check if $\vec{c} = -\vec{b}$: $-\vec{b} = -(-6\vec{i} + 8\vec{j}) = 6\vec{i} - 8\vec{j}$ 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 $\boxed{\vec{a} and \vec{c}}$.

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