Question

Of the vectors given below, the parallel vectors are $\vec{A} = 6\hat{i} + 8\hat{j}$, $\vec{B} = 210\hat{i} + 280\hat{k}$, $\vec{C} = 5.1\hat{i} + 6.8\hat{j}$, $\vec{D} = 3.6\hat{i} + 8\hat{j} + 48\hat{k}$

• A. $\vec{A}$ and $\vec{C}$
• B. $\vec{A}$ and $\vec{B}$
• C. $\vec{A}$ and $\vec{D}$
• D. $\vec{C}$ and $\vec{D}$

Hint:

To quickly check for parallel vectors, simply simplify the vectors to their smallest integer ratios. $\vec{A}$ is $2(3\hat{i} + 4\hat{j})$. $\vec{C}$ is $1.7(3\hat{i} + 4\hat{j})$. Since both reduce to the same unit direction, they are parallel.

Answer 1

The Correct Option is A

1. Analyzing $\vec{A}$ and $\vec{C}$: $\vec{A} = 6\hat{i} + 8\hat{j}$ $\vec{C} = 5.1\hat{i} + 6.8\hat{j}$ Check the ratio of components: $x$-ratio: $6 / 5.1 \approx 1.176$ $y$-ratio: $8 / 6.8 \approx 1.176$ Alternatively, check if $\vec{A} = k\vec{C}$: $6 = k(5.1) \implies k = 6/5.1 = 1.1764...$ $8 = k(6.8) \implies k = 8/6.8 = 1.1764...$ Since the scalar multiple $k$ is the same for all components, $\vec{A}$ and $\vec{C}$ are parallel.

2. Checking other options:

• $\vec{A}$ and $\vec{B}$: $\vec{A}$ has no $z$-component, while $\vec{B}$ has a large $z$-component ($280\hat{k}$) and no $y$-component. They cannot be parallel.

• $\vec{A}$ and $\vec{D}$: $\vec{D}$ has a $z$-component ($48\hat{k}$), while $\vec{A}$ does not.
Thus, only $\vec{A}$ and $\vec{C}$ are parallel vectors.