Question

The shortest distance between lines $\vec{r} = (2\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ and $\vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(2\hat{i} + \hat{j} - 5\hat{k})$ is

• A. $\frac{1}{\sqrt{5}}$ units
• B. $3$ units
• C. $\sqrt{5}$ units
• D. $2$ units

Hint:

Always glance at the direction vectors first! Notice that the first two components of $\vec{b}_1$ and $\vec{b}_2$ are identical ($2\hat{i} + \hat{j}$). This indicates that the lines share a common directional plane alignment, which simplifies the cross product greatly since the $\hat{k}$ component drops straight to zero ($2 - 2 = 0$).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Examine the direction vectors of two lines to identify shared components that simplify computing their cross product.

Step 2: Key Formula or Approach:
The cross product $\vec{b}_1 \times \vec{b}_2$ for vectors $\vec{b}_1 = a_1\hat{i} + b_1\hat{j} + c_1\hat{k}$ and $\vec{b}_2 = a_2\hat{i} + b_2\hat{j} + c_2\hat{k}$ is computed via the determinant. When matching components exist, certain terms vanish, streamlining the calculation.

Step 3: Detailed Explanation:
Observing that the first two components of both direction vectors are identical ($2\hat{i} + \hat{j}$), the cross product's $\hat{k}$ term becomes $2(1) - 1(2) = 0$. This eliminates the $\hat{k}$ component entirely, reducing the cross product to a simpler expression involving only $\hat{i}$ and $\hat{j}$ terms derived from the differing third components.

Step 4: Final Answer:
The shared components cause the $\hat{k}$ term to vanish, greatly simplifying the cross product.